# 2d Poisson Equation

This example shows how to numerically solve a Poisson's equation, compare the numerical solution with the exact solution, and refine the mesh until the solutions are close. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. Poisson's Equation in 2D We will now examine the general heat conduction equation, T t = κ∆T + q ρc. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. In it, the discrete Laplace operator takes the place of the Laplace operator. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. Recalling Lecture 13 again, we discretize this equation by using finite differences: We use an (n+1)-by-(n+1) grid on Omega = the unit square, where h=1/(n+1) is the grid spacing. The kernel of A consists of constant: Au = 0 if and only if u = c. Solving 2D Poisson on Unit Circle with Finite Elements. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. Our analysis will be in 2D. Finally, the values can be reconstructed from Eq. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. LaPlace's and Poisson's Equations. The electric field is related to the charge density by the divergence relationship. Qiqi Wang 5,667 views. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. I use center difference for the second order derivative. The code poisson_2d. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Many ways can be used to solve the Poisson equation and some are faster than others. Marty Lobdell - Study Less Study Smart - Duration: 59:56. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. 2D Poisson equation. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Elastic plates. The electric field is related to the charge density by the divergence relationship. The dotted curve (obscured) shows the analytic solution, whereas the open triangles show the finite difference solution for. ( 1 ) or the Green’s function solution as given in Eq. d = 2 Consider ˜u satisfying the wave equation in R3, launched with initial conditions invariant in the 3-direction: u˜(x1,x2,x3,0) = f˜(x1,x2,x3) = f(x1,x2),. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. Usually, is given and is sought. A video lecture on fast Poisson solvers and finite elements in two dimensions. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 3 of 16 Introduction to Scientiﬁc Computing Poisson’s Equation in 2D Michael Bader 2. The book NUMERICAL RECIPIES IN C, 2ND EDITION (by PRESS, TEUKOLSKY, VETTERLING & FLANNERY) presents a recipe for solving a discretization of 2D Poisson equation numerically by Fourier transform ("rapid solver"). a second order hyperbolic equation, the wave equation. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. 2D Poisson equations. (1) An explanation to reduce 3D problem to 2D had been described in Ref. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. Let (x,y) be a ﬁxed arbitrary point in a 2D domain D and let (ξ,η) be a variable point used for integration. (2018) Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. Our analysis will be in 2D. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. Yet another "byproduct" of my course CSE 6644 / MATH 6644. (We assume here that there is no advection of Φ by the underlying medium. Furthermore a constant right hand source term is given which equals unity. The left-hand side of this equation is a screened Poisson equation, typically stud-ied in three dimensions in physics [4]. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Solving 2D Poisson on Unit Circle with Finite Elements. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. Hence, we have solved the problem. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding ﬂux. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. This has known solution. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. 3) is to be solved in Dsubject to Dirichletboundary. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. 1D PDE, the Euler-Poisson-Darboux equation, which is satisﬁed by the integral of u over an expanding sphere. Elastic plates. These equations can be inverted, using the algorithm discussed in Sect. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. 3 Uniqueness Theorem for Poisson's Equation Consider Poisson's equation ∇2Φ = σ(x) in a volume V with surface S, subject to so-called Dirichlet boundary conditions Φ(x) = f(x) on S, where fis a given function deﬁned on the boundary. That avoids Fourier methods altogether. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). The dotted curve (obscured) shows the analytic solution, whereas the open triangles show the finite difference solution for. Yet another "byproduct" of my course CSE 6644 / MATH 6644. Solve Poisson equation on arbitrary 2D domain with RHS f and Dirichlet boundary conditions using the finite element method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. The derivation of Poisson's equation in electrostatics follows. This example shows how to numerically solve a Poisson's equation, compare the numerical solution with the exact solution, and refine the mesh until the solutions are close. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 3 of 16 Introduction to Scientiﬁc Computing Poisson’s Equation in 2D Michael Bader 2. In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. I want to use d_Helmholtz_2D(f, bd_ax, bd_bx, bd_ay, bd_by, bd_az, bd_bz, &xhandle, &yhandle, ipar, dpar, &stat)to solve the eqution with =0. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. 2D Poisson equation. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Hence, we have solved the problem. In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. Marty Lobdell - Study Less Study Smart - Duration: 59:56. Poisson on arbitrary 2D domain. It asks for f ,but I have no ideas on setting f on the boundary. In it, the discrete Laplace operator takes the place of the Laplace operator. 5 Linear Example - Poisson Equation. Yet another "byproduct" of my course CSE 6644 / MATH 6644. The book NUMERICAL RECIPIES IN C, 2ND EDITION (by PRESS, TEUKOLSKY, VETTERLING & FLANNERY) presents a recipe for solving a discretization of 2D Poisson equation numerically by Fourier transform ("rapid solver"). ( 1 ) or the Green's function solution as given in Eq. The derivation of the membrane equation depends upon the as-sumption that the membrane resists stretching (it is under tension), but does not resist bending. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. and Lin, P. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. Poisson Equation Solver with Finite Difference Method and Multigrid. That avoids Fourier methods altogether. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding ﬂux. The result is the conversion to 2D coordinates: m + p. Solve Poisson equation on arbitrary 2D domain with RHS f and Dirichlet boundary conditions using the finite element method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. A video lecture on fast Poisson solvers and finite elements in two dimensions. 4, to give the. In it, the discrete Laplace operator takes the place of the Laplace operator. Two-Dimensional Laplace and Poisson Equations. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. SI units are used and Euclidean space is assumed. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. Qiqi Wang 5,667 views. The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. 2D Poisson equations. Different source functions are considered. The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. This has known solution. Yet another "byproduct" of my course CSE 6644 / MATH 6644. That avoids Fourier methods altogether. Let (x,y) be a ﬁxed arbitrary point in a 2D domain D and let (ξ,η) be a variable point used for integration. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. 2D-Poisson equation lecture_poisson2d_draft. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Poisson’s equation can be solved for the computation of the potential V and electric field E in a [2D] region of space with fixed boundary conditions. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 1 of 16 Introduction to Scientiﬁc Computing Poisson's Equation in 2D Michael Bader 1. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Use MathJax to format equations. Marty Lobdell - Study Less Study Smart - Duration: 59:56. Homogenous neumann boundary conditions have been used. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. From a physical point of view, we have a well-deﬁned problem; say, ﬁnd the steady-. The Poisson equation arises in numerous physical contexts, including heat conduction, electrostatics, diffusion of substances, twisting of elastic rods, inviscid fluid flow, and water waves. We will consider a number of cases where fixed conditions are imposed upon. The kernel of A consists of constant: Au = 0 if and only if u = c. Poisson's Equation in 2D We will now examine the general heat conduction equation, T t = κ∆T + q ρc. 3, Myint-U & Debnath §10. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own. on Poisson's equation, with more details and elaboration. The Poisson equation arises in numerous physical contexts, including heat conduction, electrostatics, diffusion of substances, twisting of elastic rods, inviscid fluid flow, and water waves. by JARNO ELONEN ([email protected] In three-dimensional Cartesian coordinates, it takes the form. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. 3) is to be solved in Dsubject to Dirichletboundary. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 3 of 16 Introduction to Scientiﬁc Computing Poisson’s Equation in 2D Michael Bader 2. 6 Poisson equation The pressure Poisson equation, Eq. Solving a 2D Poisson equation with Neumann boundary conditions through discrete Fourier cosine transform. nst-mmii-chapte. 3) is to be solved in Dsubject to Dirichletboundary. As expected, setting λ d = 0 nulliﬁes the data term and gives us the Poisson equation. the Laplacian of u). For simplicity of presentation, we will discuss only the solution of Poisson's equation in 2D; the 3D case is analogous. Yet another "byproduct" of my course CSE 6644 / MATH 6644. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. These equations can be inverted, using the algorithm discussed in Sect. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding ﬂux. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. Poisson equation. Poisson's equation can be solved for the computation of the potential V and electric field E in a [2D] region of space with fixed boundary conditions. The left-hand side of this equation is a screened Poisson equation, typically stud-ied in three dimensions in physics [4]. (part 2); Finite Elements in 2D And so each equation comes--V is one of the. SI units are used and Euclidean space is assumed. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. by JARNO ELONEN ([email protected] LAPLACE’S EQUATION AND POISSON’S EQUATION In this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for Poisson’s equation. 4 Consider the BVP 2∇u = F in D, (4) u = f on C. In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. [2], considering an accelerator with long bunches, and assuming that the transverse motion is. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Suppose that the domain is and equation (14. Marty Lobdell - Study Less Study Smart - Duration: 59:56. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. Homogenous neumann boundary conditions have been used. Solving the 2D Poisson equation $\Delta u = x^2+y^2$ Ask Question Asked 2 years, 11 months ago. Furthermore a constant right hand source term is given which equals unity. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. FEM2D_POISSON_RECTANGLE is a C++ program which solves the 2D Poisson equation using the finite element method. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Consider the 2D Poisson equation for $1 Linear Partial Differential Equations > Second-Order Elliptic Partial Differential Equations > Poisson Equation 3. Uses a uniform mesh with (n+2)x(n+2) total 0003 % points (i. and Lin, P. The code poisson_2d. Making statements based on opinion; back them up with references or personal experience. Furthermore a constant right hand source term is given which equals unity. 2D Poisson equation. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 1 of 16 Introduction to Scientiﬁc Computing Poisson's Equation in 2D Michael Bader 1. Thus, the state variable U(x,y) satisfies:. Marty Lobdell - Study Less Study Smart - Duration: 59:56. We will consider a number of cases where fixed conditions are imposed upon. (1) An explanation to reduce 3D problem to 2D had been described in Ref. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. Furthermore a constant right hand source term is given which equals unity. Figure 63: Solution of Poisson's equation in two dimensions with simple Dirichlet boundary conditions in the -direction. The electric field is related to the charge density by the divergence relationship. 2D Poisson equation. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. LAPLACE’S EQUATION AND POISSON’S EQUATION In this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for Poisson’s equation. Solving the 2D Poisson equation$\Delta u = x^2+y^2$Ask Question Asked 2 years, 11 months ago. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Both codes, nextnano³ and Greg Snider's "1D Poisson" lead to the same results. In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. We will consider a number of cases where fixed conditions are imposed upon. These equations can be inverted, using the algorithm discussed in Sect. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. Solution to Poisson’s Equation Code: 0001 % Numerical approximation to Poisson’s equation over the square [a,b]x[a,b] with 0002 % Dirichlet boundary conditions. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 1 of 16 Introduction to Scientiﬁc Computing Poisson's Equation in 2D Michael Bader 1. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. The solution is plotted versus at. Different source functions are considered. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). The exact solution is. That avoids Fourier methods altogether. FEM2D_POISSON_RECTANGLE is a C++ program which solves the 2D Poisson equation using the finite element method. 4 Fourier solution In this section we analyze the 2D screened Poisson equation the Fourier do-main. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. The solution is plotted versus at. The kernel of A consists of constant: Au = 0 if and only if u = c. The result is the conversion to 2D coordinates: m + p. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. , , and constitute a set of uncoupled tridiagonal matrix equations (with one equation for each separate value). Finite Element Solution fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Homogenous neumann boundary conditions have been used. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. Hence, we have solved the problem. In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 4 Consider the BVP 2∇u = F in D, (4) u = f on C. The following figure shows the conduction and valence band edges as well as the Fermi level (which is constant and has the value of 0 eV) for the structure specified above. Suppose that the domain is and equation (14. The electric field is related to the charge density by the divergence relationship. Thus, the state variable U(x,y) satisfies:. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). 3) is to be solved in Dsubject to Dirichletboundary. on Poisson's equation, with more details and elaboration. The solution is plotted versus at. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. Journal of Applied Mathematics and Physics, 6, 1139-1159. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). nst-mmii-chapte. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. on Poisson's equation, with more details and elaboration. 2D Poisson-type equations can be formulated in the form of (1) ∇ 2 u = f (x, u, u, x, u, y, u, x x, u, x y, u, y y), x ∈ Ω where ∇ 2 is Laplace operator, u is a function of vector x, u,x and u,y are the first derivatives of the function, u,xx, u,xy and u,yy are the second derivatives of the function u. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. (1) An explanation to reduce 3D problem to 2D had been described in Ref. A video lecture on fast Poisson solvers and finite elements in two dimensions. We will consider a number of cases where fixed conditions are imposed upon. 4 Fourier solution In this section we analyze the 2D screened Poisson equation the Fourier do-main. a second order hyperbolic equation, the wave equation. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. The result is the conversion to 2D coordinates: m + p. Consider the 2D Poisson equation for$1 Linear Partial Differential Equations > Second-Order Elliptic Partial Differential Equations > Poisson Equation 3. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. Marty Lobdell - Study Less Study Smart - Duration: 59:56. The strategy can also be generalized to solve other 3D differential equations. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. Suppose that the domain is and equation (14. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. This Demonstration considers solutions of the Poisson elliptic partial differential equation (PDE) on a rectangular grid. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Thus, the state variable U(x,y) satisfies:. [2], considering an accelerator with long bunches, and assuming that the transverse motion is. 3 Uniqueness Theorem for Poisson's Equation Consider Poisson's equation ∇2Φ = σ(x) in a volume V with surface S, subject to so-called Dirichlet boundary conditions Φ(x) = f(x) on S, where fis a given function deﬁned on the boundary. Furthermore a constant right hand source term is given which equals unity. Use MathJax to format equations. Solving 2D Poisson on Unit Circle with Finite Elements. Recalling Lecture 13 again, we discretize this equation by using finite differences: We use an (n+1)-by-(n+1) grid on Omega = the unit square, where h=1/(n+1) is the grid spacing. The equation is named after the French mathematici. Poisson Equation Solver with Finite Difference Method and Multigrid. This example shows the application of the Poisson equation in a thermodynamic simulation. e, n x n interior grid points). Yet another "byproduct" of my course CSE 6644 / MATH 6644. The equation system consists of four points from which two are boundary points with homogeneous Dirichlet boundary conditions. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. The following figure shows the conduction and valence band edges as well as the Fermi level (which is constant and has the value of 0 eV) for the structure specified above. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. The result is the conversion to 2D coordinates: m + p. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. Suppose that the domain is and equation (14. Qiqi Wang 5,667 views. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. 3, Myint-U & Debnath §10. The exact solution is. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. The result is the conversion to 2D coordinates: m + p. on Poisson's equation, with more details and elaboration. The steps in the code are: Initialize the numerical grid; Provide an initial guess for the solution; Set the boundary values & source term; Iterate the solution until convergence; Output the solution for plotting; The code is compiled and executed via gcc poisson_2d. Hence, we have solved the problem. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. 2D Poisson equations. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. Particular solutions For the function X(x), we get the eigenvalue problem −X xx(x) = λX(x), 0 < x < 1, X(0) = X(1) = 0. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. LAPLACE’S EQUATION AND POISSON’S EQUATION In this section, we state and prove the mean value property of harmonic functions, and use it to prove the maximum principle, leading to a uniqueness result for boundary value problems for Poisson’s equation. Finite Element Solution of the 2D Poisson Equation FEM2D_POISSON_RECTANGLE , a C program which solves the 2D Poisson equation using the finite element method. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. m Benjamin Seibold Applying the 2d-curl to this equation yields applied from the left. Let (x,y) be a ﬁxed arbitrary point in a 2D domain D and let (ξ,η) be a variable point used for integration. The result is the conversion to 2D coordinates: m + p. Poisson Library uses the standard five-point finite difference approximation on this mesh to compute the approximation to the solution. Task: implement Jacobi, Gauss-Seidel and SOR-method. Two-Dimensional Laplace and Poisson Equations. In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. The equation is named after the French mathematici. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. 4, to give the. Moreover, the equation appears in numerical splitting strategies for more complicated systems of PDEs, in particular the Navier - Stokes equations. Hence, we have solved the problem. 2D-Poisson equation lecture_poisson2d_draft. As expected, setting λ d = 0 nulliﬁes the data term and gives us the Poisson equation. To show this we will next use the Finite Element Method to solve the following poisson equation over the unit circle, $$-U_{xx} -U_{yy} =4$$, where $$U_{xx}$$ is the second x derivative and $$U_{yy}$$ is the second y derivative. The steps in the code are: Initialize the numerical grid; Provide an initial guess for the solution; Set the boundary values & source term; Iterate the solution until convergence; Output the solution for plotting; The code is compiled and executed via gcc poisson_2d. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. Elastic plates. Homogenous neumann boundary conditions have been used. Thus, the state variable U(x,y) satisfies:. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Yet another "byproduct" of my course CSE 6644 / MATH 6644. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. (1) An explanation to reduce 3D problem to 2D had been described in Ref. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. and Lin, P. Task: implement Jacobi, Gauss-Seidel and SOR-method. SI units are used and Euclidean space is assumed. by JARNO ELONEN ([email protected] Poisson equation. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. The equation system consists of four points from which two are boundary points with homogeneous Dirichlet boundary conditions. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. 3 Uniqueness Theorem for Poisson's Equation Consider Poisson's equation ∇2Φ = σ(x) in a volume V with surface S, subject to so-called Dirichlet boundary conditions Φ(x) = f(x) on S, where fis a given function deﬁned on the boundary. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. We state the mean value property in terms of integral averages. Journal of Applied Mathematics and Physics, 6, 1139-1159. The left-hand side of this equation is a screened Poisson equation, typically stud-ied in three dimensions in physics [4]. 5 Linear Example - Poisson Equation. Homogenous neumann boundary conditions have been used. The steps in the code are: Initialize the numerical grid; Provide an initial guess for the solution; Set the boundary values & source term; Iterate the solution until convergence; Output the solution for plotting; The code is compiled and executed via gcc poisson_2d. The solution is plotted versus at. 2D Poisson equation. Elastic plates. We discretize this equation by using finite differences: We use an (n+1)-by-(n+1) grid on Omega = the unit square, where h=1/(n+1) is the grid spacing. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. FINITE DIFFERENCE METHODS FOR POISSON EQUATION 5 Similar techniques will be used to deal with other corner points. Poisson Equation Solver with Finite Difference Method and Multigrid. These equations can be inverted, using the algorithm discussed in Sect. 3) is to be solved in Dsubject to Dirichletboundary. Thus, solving the Poisson equations for P and Q, as well as solving implicitly for the viscosity terms in U and V, yields. From a physical point of view, we have a well-deﬁned problem; say, ﬁnd the steady-. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Two-Dimensional Laplace and Poisson Equations In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first compute the solution to the steady state equation. 4 Consider the BVP 2∇u = F in D, (4) u = f on C. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Different source functions are considered. The code poisson_2d. 2D Poisson-type equations can be formulated in the form of (1) ∇ 2 u = f (x, u, u, x, u, y, u, x x, u, x y, u, y y), x ∈ Ω where ∇ 2 is Laplace operator, u is a function of vector x, u,x and u,y are the first derivatives of the function, u,xx, u,xy and u,yy are the second derivatives of the function u. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). the full, 2D vorticity equation, not just the linear approximation. 1 Introduction Many problems in applied mathematics lead to a partial di erential equation of the form 2aru+ bru+ cu= f in. I use center difference for the second order derivative. nst-mmii-chapte. Solving a 2D Poisson equation with Neumann boundary conditions through discrete Fourier cosine transform. Qiqi Wang 5,667 views. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and biharmonic equations. Suppose that the domain is and equation (14. This has known solution. SI units are used and Euclidean space is assumed. Statement of the equation. Poisson equation. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own. (We assume here that there is no advection of Φ by the underlying medium. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. Moreover, the equation appears in numerical splitting strategies for more complicated systems of PDEs, in particular the Navier - Stokes equations. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. A partial semi-coarsening multigrid method is developed to solve 3D Poisson equation. Find optimal relaxation parameter for SOR-method. 1 Note that the Gaussian solution corresponds to a vorticity distribution that depends only on the radial variable. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. The book NUMERICAL RECIPIES IN C, 2ND EDITION (by PRESS, TEUKOLSKY, VETTERLING & FLANNERY) presents a recipe for solving a discretization of 2D Poisson equation numerically by Fourier transform ("rapid solver"). In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Poisson equation. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. Poisson Library uses the standard five-point finite difference approximation on this mesh to compute the approximation to the solution. Finally, the values can be reconstructed from Eq. The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. We will consider a number of cases where fixed conditions are imposed upon. I use center difference for the second order derivative. Multigrid This GPU based script draws u i,n/4 cross-section after multigrid V-cycle with the reduction level = 6 and "deep" relaxation iterations 2rel. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Finite Element Solution fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. Moreover, the equation appears in numerical splitting strategies for more complicated systems of PDEs, in particular the Navier - Stokes equations. The electric field is related to the charge density by the divergence relationship. fem2d_poisson_rectangle, a MATLAB program which solves the 2D Poisson equation using the finite element method, and quadratic basis functions. 1 $\begingroup$ Consider the 2D Poisson equation. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. 2D Poisson equations. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. nst-mmii-chapte. m Benjamin Seibold Applying the 2d-curl to this equation yields applied from the left. the steady-state diﬀusion is governed by Poisson’s equation in the form ∇2Φ = − S(x) k. Qiqi Wang 5,667 views. A partial semi-coarsening multigrid method is developed to solve 3D Poisson equation. For simplicity of presentation, we will discuss only the solution of Poisson's equation in 2D; the 3D case is analogous. Hence, we have solved the problem. I use center difference for the second order derivative. 4, to give the. the steady-state diﬀusion is governed by Poisson’s equation in the form ∇2Φ = − S(x) k. Solve Poisson equation on arbitrary 2D domain with RHS f and Dirichlet boundary conditions using the finite element method. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary, and the Poisson equation applied inside. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. The code poisson_2d. m Benjamin Seibold Applying the 2d-curl to this equation yields applied from the left. Our analysis will be in 2D. Two-Dimensional Laplace and Poisson Equations. 1 $\begingroup$ Consider the 2D Poisson equation. Poisson's Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 1 of 16 Introduction to Scientiﬁc Computing Poisson's Equation in 2D Michael Bader 1. Poisson Equation Solver with Finite Difference Method and Multigrid. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. Poisson equation. ( 1 ) or the Green’s function solution as given in Eq. the steady-state diﬀusion is governed by Poisson’s equation in the form ∇2Φ = − S(x) k. The Two-Dimensional Poisson Equation in Cylindrical Symmetry The 2D PE in cylindrical coordinates with imposed rotational symmetry about the z axis maybe obtained by introducing a restricted spatial dependence into the PE in Eq. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. It asks for f ,but I have no ideas on setting f on the boundary. In it, the discrete Laplace operator takes the place of the Laplace operator. Elastic plates. A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains mit18086 navierstokes. Solving 2D Poisson on Unit Circle with Finite Elements. Let Φ(x) be the concentration of solute at the point x, and F(x) = −k∇Φ be the corresponding ﬂux. Uses a uniform mesh with (n+2)x(n+2) total 0003 % points (i. fem2d_poisson_sparse, a program which uses the finite element method to solve Poisson's equation on an arbitrary triangulated region in 2D; (This is a version of fem2d_poisson which replaces the banded storage and direct solver by a sparse storage format and an iterative solver. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. Suppose that the domain is and equation (14. In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. The electric field is related to the charge density by the divergence relationship. Making statements based on opinion; back them up with references or personal experience. Poisson Equation Solver with Finite Difference Method and Multigrid. This is often written as: where is the Laplace operator and is a scalar function. The 2D Poisson equation is solved in an iterative manner (number of iterations is to be specified) on a square 2x2 domain using the standard 5-point stencil. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. FEM2D_POISSON_RECTANGLE, a C program which solves the 2D Poisson equation using the finite element method. Poisson Library uses the standard five-point finite difference approximation on this mesh to compute the approximation to the solution. (1) An explanation to reduce 3D problem to 2D had been described in Ref. Let r be the distance from (x,y) to (ξ,η),. 6 Poisson equation The pressure Poisson equation, Eq. Two-Dimensional Laplace and Poisson Equations. In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. Solving the 2D Poisson equation $\Delta u = x^2+y^2$ Ask Question Asked 2 years, 11 months ago. 2 Solution of Laplace and Poisson equation Ref: Guenther & Lee, §5. Figure 65: Solution of Poisson's equation in two dimensions with simple Neumann boundary conditions in the -direction. 2 Inserting this into the Biot-Savart law yields a purely tangential velocity eld. This example shows the application of the Poisson equation in a thermodynamic simulation. A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. The Poisson equation on a unit disk with zero Dirichlet boundary condition can be written as -Δ u = 1 in Ω, u = 0 on δ Ω, where Ω is the unit disk. c implements the above scheme. the Laplacian of u). For simplicity of presentation, we will discuss only the solution of Poisson's equation in 2D; the 3D case is analogous. Marty Lobdell - Study Less Study Smart - Duration: 59:56. Furthermore a constant right hand source term is given which equals unity. (2018) Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). e, n x n interior grid points). (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Poisson Equation Solver with Finite Difference Method and Multigrid. nst-mmii-chapte. Both codes, nextnano³ and Greg Snider's "1D Poisson" lead to the same results. 2D Poisson equations. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. Elastic plates. Poisson’s Equation in 2D Analytic Solutions A Finite Difference A Linear System of Direct Solution of the LSE Classiﬁcation of PDE Page 3 of 16 Introduction to Scientiﬁc Computing Poisson’s Equation in 2D Michael Bader 2. Two-Dimensional Laplace and Poisson Equations. Qiqi Wang 5,667 views. The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. 1D PDE, the Euler-Poisson-Darboux equation, which is satisﬁed by the integral of u over an expanding sphere. In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. pro This is a draft IDL-program to solve the Poisson-equation for provide charge distribution. I use center difference for the second order derivative. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. This has known solution. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). Particular solutions For the function X(x), we get the eigenvalue problem −X xx(x) = λX(x), 0 < x < 1, X(0) = X(1) = 0. and Lin, P. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Usually, is given and is sought. Finite Volume model in 2D Poisson Equation This page has links to MATLAB code and documentation for the finite volume solution to the two-dimensional Poisson equation where is the scalar field variable, is a volumetric source term, and and are the Cartesian coordinates. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Solving 2D Poisson on Unit Circle with Finite Elements. Our analysis will be in 2D. Qiqi Wang 5,667 views. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. The derivation of Poisson's equation in electrostatics follows. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. (We assume here that there is no advection of Φ by the underlying medium. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. SI units are used and Euclidean space is assumed. Journal of Applied Mathematics and Physics, 6, 1139-1159. LaPlace's and Poisson's Equations. Qiqi Wang 5,667 views. Use MathJax to format equations. FINITE DIFFERENCE METHODS FOR POISSON EQUATION 5 Similar techniques will be used to deal with other corner points. Let r be the distance from (x,y) to (ξ,η),. (part 2); Finite Elements in 2D And so each equation comes--V is one of the. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. 6 is used to create a velocity eld that satis es the continuity equation and is incompressible. 4, to give the. ( 1 ) or the Green’s function solution as given in Eq. Making statements based on opinion; back them up with references or personal experience. Research highlights The full-coarsening multigrid method employed to solve 2D Poisson equation in reference is generalized to 3D. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. This has known solution. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. on Poisson's equation, with more details and elaboration. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Viewed 392 times 1. 2D Poisson equation. e, n x n interior grid points). In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. nst-mmii-chapte. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. Thus, the state variable U(x,y) satisfies:. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. , , and constitute a set of uncoupled tridiagonal matrix equations (with one equation for each separate value). The diﬀusion equation for a solute can be derived as follows. 2D Poisson equation. The solution is plotted versus at. This is often written as: where is the Laplace operator and is a scalar function. Solving 2D Poisson on Unit Circle with Finite Elements. From a physical point of view, we have a well-deﬁned problem; say, ﬁnd the steady-. a second order hyperbolic equation, the wave equation. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-deﬁnite (see Exercise 2). If the membrane is in steady state, the displacement satis es the Poisson equation u= f;~ f= f=k. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i.
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# Why does at least two intervals overlap in an uncountable family of intervals?

1. Prove that there are uncountably many intervals $(a,b)$ in $\mathbb{R}, a\neq b$.
2. Assume $X$ be an uncountable family of intervals. Show that there exists at least two intervals in this family that overlap.

First was not difficult. I used the arguments similar to Cantor's Diagonal Argument (used to show $\mathbb{R}$ is uncountable.)

My attempt for 2: Assume $X$ be an uncountable family of pairwise disjoint intervals, i.e. $(a_i,b_i) \cap (a_j,b_j) = \emptyset, \quad \forall i\neq j\in I$. We know there exists a rational number in each of these intervals. This implies there are uncountably many rational numbers. Contradiction, since $\mathbb{Q}$ is countable. Thus, $X$ must have at least two intervals that overlap. $\blacksquare$

Is there any problem with this reasoning?

• First is a consequence of $(a,b)\mapsto b-a$ being a surjection. – Git Gud Sep 29 '13 at 11:33
• So would it be wrong to argue the way I did? – math Sep 29 '13 at 11:35
• Probably not, but it seems to me too complicated to prove what you want to prove. – Git Gud Sep 29 '13 at 11:36
• Looks fine to me. What about the argument made you feel uncertain? – Callus - Reinstate Monica Sep 29 '13 at 11:38
• @Callus, are you asking me or Git Gud? – math Sep 29 '13 at 11:45

For (1), you certainly could use a diagonal argument directly to prove that there is no surjection from $\mathbb{N}$ onto the set of intervals, or as Git Gud points out you could instead use the existence of a surjection from the set of intervals to $\mathbb{R}$ and then appeal to the nonexistence of a surjection $\mathbb{N} \to \mathbb{R}$. It is common to use Cantor's theorem on the uncountability of the reals as a "black box" in this way.
Your proof for (2) is perfectly fine. You could also get the contradiction by showing that $X$ is countable after all, rather than by showing that $\mathbb{Q}$ is uncountable, but this choice is just a matter of taste.
• For 1), why not just say that $(0,x)$ is an interval for any $x>0$, giving us directly an injection of $\mathbb R^+$ into the set of intervals? (Even simpler than the route with surjections.) – Andrés E. Caicedo Sep 29 '13 at 16:45
• A third method would simply use the surjection from intervals to reals given by $(a,b) \mapsto a$. – Trevor Wilson Sep 29 '13 at 16:53
• @Asaf I think that depends on whether or not it starts with "let $X$ be an uncountable family of pairwise disjoint intervals" like the OP's proof does. – Trevor Wilson Sep 29 '13 at 17:18


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Now, let's try to do the opposite, i.e., calculate the K sp from the solubility of a salt. The solubility of insoluble substances can be decreased by the presence of a common ion. Performance & security by Cloudflare, Please complete the security check to access. The solubility product of $\ce{AgCl}$ in water is given as $1.8\cdot10^{-10}$ Determine the solubility of $\ce{AgCl}$ in pure water and in a $0.25~\mathrm{M}$ $\ce{MgSO4}$ solution. fjs.parentNode.insertBefore(js, fjs); (function(d, s, id) { Your IP: 37.252.96.253 Apart from being the largest NEET community, EduRev has the largest solved Calculate the solubility of AgCl in a 0.1 M NaCl solution. Explanation: Jamunaakailash … L6 : Solubility in water - Sorting Materials, Science, Class 6, Solubility and Solubility product - Ionic Equilibrium, Solubility and solubility product - Ionic Equilibrium. is done on EduRev Study Group by NEET Students. Calculate the ratio of solubility of agcl in 0.1m agno3 and in pure water - 6820732 1. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. if (d.getElementById(id)) return; c. If 50.0 ml of 0.10M AgNO3 are added to 50.0 ml of 0.15M NaCl will a precipitate be formed? For the precipitation of BaSO4 from the solution of BaCl2, the conc. Calculate the solubility of AgCl (K sp = 1.8 x10-10) in pure water. are solved by group of students and teacher of NEET, which is also the largest student Does the solubility of this salt increase or decrease, compared to its solubility in pure water? js.src = "//connect.facebook.net/en_US/sdk.js#xfbml=1&version=v2.10"; Join now. First, write the BALANCED REACTION: Next, set up the SOLUBILITY PRODUCT EQUILIBRIUM EXPRESSION: It is given in the problem that the solubility of AgCl is 1.3 x 10-5. Want to see the step-by-step answer? … Calculate solubility of AgCl in a) pure water b) 0.1 M AgNO 3 c) 0.01 M NaCl Solution ) a) Solubility of AgCl in pure water-Solubility product ‘K sp ‘ = 1.5 x 10-10. 1. Cloudflare Ray ID: 5f8635b20c77ff38 Check out a … Question 2)Solubility product of AgCl is 1.5 x 10-10 at 25 0 C . If the answer is not available please wait for a while and a community member will probably answer this over here on EduRev! Again, these concentrations give the solubility. Present in silver chloride are silver ions (A g +) and chloride ions (C l −). To calculate the ratio of solubility of AgCl in 0.1M AgNO^3 and in pure water what you need to put in the back of your mind is that water has a constant concentration of 1.0 x 10^-7M. of Mg(OH)2 = 24 + (16 + 1)2 = 24 + 34  = 58, S in mole /litre = ( S in gm /litre) / mol.wt.= 9.57 x 10 -3 / 58, S is less than 0.02 , so 4S3 is neglected, S in gm / litre = 1.49 x 10 -5 x 58 = 86.42 x 10-5. because S <<< 0.1 and solubility of AgCl decreases in presence of AgNO3 due to common ion effect(common ion is Ag+).Therefore. By continuing, I agree that I am at least 13 years old and have read and a. Then [Cl-] = x, and substituting in the Ksp expression we have (x)(x) = 1.8 x 10 ^-10. S = √ K sp = √ 1.5x 10 -10 The molar solubility of AgCl = [Ag+] in pure water . Calculate the solubility of AgCl in a 0.20 M AgNO3 solution. Ask your question. community of NEET. The Questions and Answer. check_circle Expert Answer. K sp = S 2. A g C l is not soluble in water. x = 1.342 x 10 ^-05 moles dm^-3 = molar solubility of AgCl. Again, these concentrations give the solubility. Question bank for NEET. You may need to download version 2.0 now from the Chrome Web Store. Log in. To calculate the ratio of solubility of AgCl in 0.1M AgNO^3 and in pure water what you need to put in the back of your mind is that water has a constant concentration of 1.0 x 10^-7M. Join now. }(document, 'script', 'facebook-jssdk')); Online Chemistry tutorial that deals with Chemistry and Chemistry Concept. • Click hereto get an answer to your question  The solubility product of AgCl in water is 1.5 × 10^-10 . d. Calculate the solubility of AgCl in a 0.60 M NH 3 solution. Solubility of AgCl will be minimum in _____. varunnair3842 varunnair3842 26.11.2018 Chemistry Secondary School Calculate the ratio of solubility of agcl in 0.1m agno3 and in pure water 2 See answers shauryakesarwani21 shauryakesarwani21 Answer: 1.34 X 10^-4. Calculate the solubility of AgCl in a 0.20 M AgNO3 solution. • Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. soon. js = d.createElement(s); js.id = id; Log in. Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? agree to the. because S <<< 0.01 and solubility of AgCl decreases in presence of  NaCl due to common ion effect(common ion is Cl–).Therefore, ionic product of [Ba++] [SO4– –] > K sp of BaSO4. See Answer . Answers of Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Question. So with that information, we can say that the solubility of AgCl in 0.1M AgNO^3 is given as 0.1 M. So to determine the ratio of solubility of AgCl in 0.1M AgNO^3 and water all we need to do is; This discussion on Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water?


# Find all real numbers $a_1, a_2, a_3, b_1, b_2, b_3$.

Find all real numbers $$a_1, a_2, a_3, b_1, b_2, b_3$$ such that for every $$i\in \lbrace 1, 2, 3 \rbrace$$ numbers $$a_{i+1}, b_{i+1}$$ are distinct roots of equation $$x^2+a_ix+b_i=0$$ (suppose $$a_4=a_1$$ and $$b_4=b_1$$).

There are many ways to do it but I've really wanted to finish the following idea:

From Vieta's formulas we get:

\begin{align} \begin{cases} a_1+b_1=-a_3 \ \ \ \ \ \ \ \ (a) \\a_2+b_2=-a_1\ \ \ \ \ \ \ \ (b)\\a_3+b_3=-a_2\ \ \ \ \ \ \ \ (c)\\a_1b_1=b_3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (d)\\a_2b_2=b_1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (e)\\a_3b_3=b_2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (f)\end{cases} \end{align} First we notice that each $$b_i$$ is nonzero. Indeed, suppose $$b_1=0$$. Then from (d) and (f) we deduce that $$b_3=0$$ and $$b_2=0$$, so from (a), (b), (c) we get $$a_1=-a_3=-(-a_2)=-(-(-a_1))=-a_1$$, hence $$a_1=0$$ which is impossible.

Now, from (a), (b), (c), (d), (e), (f) we obtain: \begin{align} \begin{cases} a_1+b_1-a_1b_1=-a_3-b_3 \ \ \ \ \ \ \ \ \\a_2+b_2-a_2b_2=-a_1-b_1\ \ \ \ \ \ \ \ \\a_3+b_3-a_3b_3=-a_2-b_2\end{cases}, \end{align} so: \begin{align} \begin{cases} (b_1-1)(a_1-1)=1-a_2 \ \ \ \ \ \ \ \ \\(b_2-1)(a_2-1)=1-a_3\ \ \ \ \ \ \ \ \\(b_3-1)(a_3-1)=1-a_1\end{cases}. \end{align} Therefore: \begin{align*} (b_1-1)(b_2-1)(b_3-1)(a_1-1)(a_2-1)(a_3-1)=(1-a_1)(1-a_2)(1-a_3), \end{align*} which implies: \begin{align*} \bigl((a_1-1)(a_2-1)(a_3-1)\bigr)\bigl((b_1-1)(b_2-1)(b_3-1)+1\bigr)=0. \end{align*}

I got stuck here. Is it possible to prove that in this case $$b_i=0$$ is the only solution to equation $$(b_1-1)(b_2-1)(b_3-1)=-1$$ or maybe get contradiction in some other way? If so, we can assume that $$a_1=1$$ and from here we can easily show that also $$a_2=a_3=1$$, so $$b_1=b_2=b_3=-2$$.

Since $$(b_1-1)(b_2-1)(b_3-1)>-1$$ for every $$b_i>0$$ and $$(b_1-1)(b_2-1)(b_3-1)<-1$$ for every $$b_i<0$$, it suffices to prove that the signs of $$b_1, b_2, b_3$$ can't be different but I don't know how to do it. I also found out that $$(b_1+1)^2+(b_2+1)^2+(b_3+1)^2=3$$, so $$b_i\in [-\sqrt{3}-1, \sqrt{3}-1]$$ but I don't know if we can use it somehow.

Well, here is a way to proceed after you note none of the $$b_i$$ are zero. Hints:
1) Multiplying the last three equations, $$(d)\times (e)\times (f)$$ gives $$a_1a_2a_3=1$$.
2) Now, multiplying just any two among those, e.g. $$(d)\times (e)$$ and using the result 1) above gives $$a_2b_2=a_3b_3$$, by symmetry and simplification using $$(d), (e), (f)$$ gives $$b_i = b$$, for some non-zero constant $$b$$, and hence $$a_i=1$$.
3) Now it is easy to conclude $$b=-2$$ from any of the first three equations. Hence $$(a_i, b_i)=(1, -2)$$.
• Yes I've also done this but I was specifically asking about the idea with equation $(b_1-1)(b_2-1)(b_3-1)=-1$. Does it mean it's not possible to finish that solution? – glopf Jun 22 at 23:02
• By itself, $(b_1-1)(b_2-1)(b_3-1)=-1$ does not allow one to conclude anything useful, you have to use some among equations $(a)-(f)$ in addition to draw a contradiction. Then why not solve $(a)-(f)$, which is what is really needed anyway? – Macavity Jun 23 at 16:57


Smallest number of points on plane that guarantees existence of a small angle

What is the smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $18^\circ$?

It is clear that $n>9$, since the vertices of a regular 9-gon is a counterexample. One can prove using pigeonhole principle that $n\le 11$. Take a edge of the convex hull of points. All the points lie to one side of this line. cut the half-plane into 10 slices of $18^\circ$ each. There can't be any points in the first and last slice. Thus, by pigeonhole principle some slice contains more than one point. So we have an angle of size at most $18^\circ$.

Now, for $n=10$, I can not come up with a counterexample nor a proof of correctness. Any ideas?

Hmm this appears pretty complicated. I hope there is a simpler solution.

Theorem. For any $n\ge3$ points on a 2D plane, we can always find 3 points $A,B,C$ such that $\angle ABC\le\frac{180°}n$. (OP's solution follows with $n=10$.)

Proof. Again we take an edge of the convex hull and all points lie on the other side of this edge. Let's say points $A$ and $B$ are on this edge.

Partition the half-plane into $n$ regions about point $A$. Name each region $A_1,\dotsc,A_n$, where a point $X$ inside $A_i$ means $\frac{180°}n(i-1)<\angle XAB\le \frac{180°}n\cdot i$.

Do the same about point $B$. This will create $n$ other region $B_1,\dotsc,B_n$, where point $X$ in $B_j$ means $\frac{180°}n(j-1)<\angle XBA\le \frac{180°}n\cdot j$.

If two points $X,Y$ can be found in the same region $A_i$, then we also have $\angle XAY \le \frac{180°}n$. That means only one point can exist in each of the region $A_i$ (there are $n-2$ points and $n-1$ valid regions). Same for $B_j$.

$A_i$ and $B_j$ may overlap, though only when $i+j< n+2$ (a):

\begin{array}{c|ccccccc} & A_2 & A_3 & A_4 & \cdots & A_{n-2} & A_{n-1} & A_n \\ \hline B_2 & ✓ & ✓ & ✓ & \cdots & ✓ & \color{red}{✓} & ✗ \\ B_3 & ✓ & ✓ & ✓ & \cdots & \color{red}{✓} & ✗ & ✗ \\ B_4 & ✓ & ✓ & ✓ & \cdots & ✗ & ✗ & ✗ \\ \vdots & \vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\ B_{n-2} & ✓ & \color{red}{✓} & ✗ & \cdots & ✗ & ✗ & ✗ \\ B_{n-1} & \color{red}{✓} & ✗ & ✗ & \cdots & ✗ & ✗ & ✗ \\ B_n & ✗ & ✗ & ✗ & \cdots & ✗ & ✗ & ✗ \\ \end{array}

we immediately see that the points can only exist in the overlapping regions $A_i B_{n+1-i}$, to ensure no two point occupy the same column ($A_i$) or same row ($B_j$).

So for any $X$ in $A_i B_{n+1-i}$, we have:

\begin{align} \angle XAB &> \frac{180°}n(i-1) \\ \angle XBA &> \frac{180°}n(n-i) \\ \end{align}

For the triangle $\triangle ABX$, we need $\angle XAB + \angle XBA + \angle AXB = 180°$, thus

$$\angle AXB < 180° - \frac{180°}n(i-1) - \frac{180°}n(n-i) = \frac{180°}n.$$

Note:

(a) if point $X$ exists in both $A_i$ and $B_j$ with $i+j\ge n+2$ then $$\angle XAB + \angle XBA > \frac{180°}n(i+j-2) \ge 180°,$$ which is impossible.

Suppose there are $10$ points. Let us construct the convex hull of these points. It has at most $10$ sides. This means it must have an angle that is at most $180° - 360°/10 = 144°$. Let us choose the vertex corresponding to such angle as point $A$, and its 2 neighboring points along the convex hull as points $B$ and $C$. In other words, we have situation pictured here:

Now, from $7$ points inside the angle $\angle CAB$ let us draw 7 lines to point $A$. This will divide the angle into 8 smaller angles, and one of them will be at most $144°/8 = 18°$.

Therefore, since there is a counterexample for $n=9$ (a regular 9-gon), the answer is

$$n=10$$

$$GENERALIZATION$$

The smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $180°/k$ is $$n=k$$.

The proof is virtually the same as for the case $k=10$.

• @Untitled can you perhaps review two answers and see if they solved the problem? :) – VividD May 2 '15 at 8:40


# Proving formula for product of first n odd numbers

I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$):

$$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$

How can I prove that it holds (or find a counter-example)?

• This sounds like homework. Did you try induction? – cardinal Oct 2 '11 at 1:20
• @Pedro: Do you mean $$\left(\prod_{i=1}^n2i\right) - 1$$ (which is what you wrote) or $$\prod_{i=1}^n(2i-1)$$(which is what your title suggests)? – Arturo Magidin Oct 2 '11 at 1:25
• It is not, I stumbled upon this formula by accident and was wondering. The proof was fairly simple, I suppose I should have thought more about it – Pedro Oct 2 '11 at 1:28
• @ArturoMagidin: I meant the latter, sorry for the ambiguity – Pedro Oct 2 '11 at 1:30
• As a note: what you have is sometimes termed as the "double factorial". – J. M. is a poor mathematician Oct 2 '11 at 9:33

The idea is to "complete the factorials":

$$1\cdot 3 \cdot 5 \cdots (2n-1) = \frac{ 1 \cdot 2 \cdot 3 \cdot 4 \cdots (2n-1)\cdot (2n) }{2\cdot 4 \cdot 6 \cdots (2n)}$$

Now take out the factor of $2$ from each term in the denominator:

$$= \frac{ (2n)! }{2^n \left( 1\cdot 2 \cdot 3 \cdots n \right)} = \frac{(2n)!}{2^n n!}$$

A mathematician may object that there is a small gray area about what exactly happens between those ellipses, so for a completely rigorous proof one would take my post and incorporate it into a proof by induction.

For the induction argument, \begin{align*} \prod_{i=1}^{n+1}(2i-1)&=\left(\prod_{i=1}^n(2i-1)\right)(2n+1)\\ &= \frac{(2n)!(2n+1)}{2^n n!} \end{align*} by the induction hypothesis. Now multiply that last fraction by a carefully chosen expression of the form $\dfrac{a}a$ to get the desired result.

$$\Pi_{k=1}^n(2k-1) = \Pi_{k=1}^n(2k-1) \frac{\Pi_{k=1}^n(2k)}{\Pi_{k=1}^n(2k)} =\frac{\Pi_{k=1}^{2n}k}{2^n\Pi_{k=1}^n k} = \frac{(2n)!}{2^n n!}$$

• This is the same answer as the one given four years ago by Ragib, only with fewer explanations. It's good that you want to help by answering questions, but maybe you could consider answering some that haven't been answered yet? – mrf Aug 31 '15 at 20:42


# help:how to slove this integral

Note by Abhinavyukth Suresh
9 months, 2 weeks ago

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Simply use the Gamma Function. After the substitution $ar^{4}=t$, the integral becomes $\frac{1}{4a^{\frac{3}{4}}}\int_{0}^{\infty}t^{-\frac{1}{4}}e^{-t}dt$. The answer comes out to be $\frac{\Gamma\left(\frac{3}{4}\right)}{4a^{\frac{3}{4}}}$

- 9 months, 2 weeks ago

You could ask @Aruna Yumlembam - he's an expert on Gamma Functions - he likes them @Abhinavyukth Suresh

- 9 months, 2 weeks ago

- 9 months, 2 weeks ago

He'll probably show the entire proof - see his notes. @Aaghaz Mahajan

- 9 months, 2 weeks ago

There is no "proof" neede here. Simply substitute $ar^{4}=t$ and then the answer follows

- 9 months, 2 weeks ago

Ok. But I believe he should come. He's good at this stuff. I've read his notes - and gave me a interesting infinite series / function in one of my notes.

- 9 months, 2 weeks ago

Ok. Although i dont see what else might be needed in the proof.

- 9 months, 2 weeks ago

Maybe his insight into the Gamma Function?

- 9 months, 2 weeks ago

Also, you could learn a thing or two from him...

Not saying you're bad or anything.

- 9 months, 2 weeks ago

so, isn't there any other method than using gamma function to solve this integral?

- 9 months, 2 weeks ago

The proof given by Mr.Aaghaz is correct and mine is same too .Yet using this very idea we can prove this result, $\frac{\Gamma(1/n)}{n}\zeta(1/n)=\int_0^\infty\frac{1}{e^{x^n}-1}dx$, giving us, $\frac{1}{n}\int_0^\infty\frac{x^{1/n-1}}{e^x-1}dx=\int_0^\infty\frac{1}{e^{x^n}-1}dx$as the result.

- 9 months, 2 weeks ago

Yeah we can prove the identity by summing an infinite GP too. Also these types of integrals are known as Bose Einstein Integrals

- 9 months, 2 weeks ago

Thanks a lot.But can you please give me a your solution to your problem How is this ??!!

- 9 months, 2 weeks ago

Sure. Observe that $\sum_{n=1}^{\infty}e^{-nx}\ =\ \frac{1}{e^{x}-1}$

Using this, we have $\int_{0}^{\infty}\frac{x^{t-1}}{e^{x}-1}dx=\int_{0}^{\infty}\left(\sum_{n=1}^{\infty}x^{t-1}e^{-nx}\right)dx$

Swapping the integral and summation , and using the identity $\int_{0}^{\infty}x^{a}e^{-bx}dx\ =\ \frac{a!}{b^{a+1}}$ we will arrive at the answer.

- 9 months, 2 weeks ago

Mr. Yajat Shamji,if people provides a solution to any problem you must try and appreciate it and not discourage them.Please don't repeat such acts.

- 9 months, 2 weeks ago

I'm not. I.. was thinking of you and I read your profile so I thought I could bring you over. After all, you like to contribute, right?

Also, I wasn't discouraging @Aaghaz Mahajan's solution, I..

- 9 months, 2 weeks ago

- 9 months, 2 weeks ago

@Aruna Yumlembam - @Abhinavyukth Suresh needs your help on solving this integral - needs the Gamma Function and a full proof.


How can you show by hand that $e^{-1/e} < \ln(2)$?

By chance I noticed that $$e^{-1/e}$$ and $$\ln(2)$$ both have decimal representation $$0.69\!\ldots$$, and it got me wondering how one could possibly determine which was larger without using a calculator.

For example, how might Euler have determined that these numbers are different?

• Probably the easiest approach is to show that $$\frac{1}{e}+\ln(\ln(2))$$ is positive. – Peter Aug 29 at 18:06
• What do exactly mean the dots? Namely, do you want to prove that $.692200<e^{-1/e}<.692201$? Or what you want to prove is the inequality $e^{-1/e}<\ln 2$? – ajotatxe Aug 29 at 18:07
• @ajotatxe The challenge is to show $$e^{-1/e}<\ln(2)$$ – Peter Aug 29 at 18:08
• I'm happy to change the title. I was just trying to point out how remarkably close they are. – Kevin Beanland Aug 29 at 18:12
• Thanks. I just changed it. – Kevin Beanland Aug 29 at 18:14

Here's a method that only uses rational numbers as bounds, keeping to small denominators where possible. It is elementary, but finding sufficiently good bounds requires some experimentation or at least luck.

We instead establish the inequality $$-\log \log 2 < \frac{1}{e}$$ of positive numbers, which we can see to be equivalent by taking the logarithm of and then negating both sides. Our strategy for establishing this latter inequality (which is equivalent to yet another inequality suggested in the comments) is to use power series to produce an upper bound for the less hand side and a larger lower bound for the right-hand side. To estimate logarithms, we use the identity $$\log x = 2 \operatorname{artanh} \left(\frac{x - 1}{x + 1}\right) ,$$ which yields faster-converging series and so lets us use fewer terms for our estimate.

Firstly, $$\log 2 = 2 \operatorname{artanh} \frac{1}{3} .$$ Then, since the power series $$\operatorname{artanh} u = u + \frac{1}{3} u^3 + \frac{1}{5} u^5 + \cdots$$ has nonnegative coefficients, for $$0 < u < 1$$ any truncation thereof is a lower bound for the series, and in particular $$\log 2 = 2 \operatorname{artanh} \frac{1}{3} > 2\left[\left(\frac{1}{3}\right) + \frac{1}{3} \left(\frac{1}{3}\right)^3 + \frac{1}{5} \left(\frac{1}{3}\right)^5\right] = \frac{842}{1215} .$$ We'll use the same power series to produce an upper bound for $$-\log \log 2$$, but since we're nominally computing by hand and want to avoid computing powers of a rational number with large numerator and denominator, we'll content ourselves with a weaker rational lower bound for which computing powers is faster: Cross-multiplying shows that $$\log 2 > \frac{842}{1215} > \frac{9}{13}$$ and so $$-\log \log 2 < -\log \frac{9}{13} = 2 \operatorname{artanh} \frac{2}{11} .$$ This time, we want an upper bound for $$2 \operatorname{artanh} \frac{2}{11}$$. When $$0 < u < 1$$ we have $$\operatorname{artanh} u = \sum_{k = 0}^\infty \frac{1}{2 k + 1} u^{2 k + 1} < u + \frac{1}{3} u^3 \sum_{k = 0}^\infty u^{2k} = u + \frac{u^3}{3 (1 - u^2)},$$ and evaluating at $$u = \frac{2}{11}$$ gives $$2 \operatorname{artanh} \frac{2}{11} < \frac{1420}{3861} < \frac{32}{87} .$$ On the other hand, the series for $$\exp(-x)$$ alternates, giving the estimate $$\frac{1}{e} = \sum_{k=0}^\infty \frac{(-1)^k}{k!} > \sum_{k=0}^7 \frac{(-1)^k}{k!} = \frac{103}{280} .$$ Combining our bounds gives the desired inequality $$-\log \log 2 < \frac{32}{87} < \frac{103}{280} < \frac{1}{e} .$$

I'm assuming (partly because he did create tables) that Euler would know the values of $$e$$ and $$\ln(2)$$ to at least 4 decimal places, and your "by hand" guy should also know those. Explicitly calculating $$e^{-1/e}$$, however, hast to be declared "too hard"; otherwise the problem becomes a triviality.

The following reasoning is well within Euler's knowledge base, and requires no really difficult arithmetic:

Express $$e^{-1/e}$$ as its Taylor series, and expand each term as a Taylor series for $$e^{-n}$$: $$\begin{array}{clllllc}e^{-1/e}=&&+1 &&- \frac1{e}&+\frac1{2e^2}&-\frac1{6e^3} &+\cdots \\ e^{-1/e}=& &&-1&+1&-\frac12&+\frac16&-\cdots \\&&&+\frac12&-\frac22 & +\frac{2^2}{2!\cdot 2}&-\frac{2^3}{3!\cdot 2}&+\cdots \\&&&-\frac16&+\frac36 & -\frac{3^2}{3!\cdot 6}&+\frac{3^3}{3!\cdot 6}&-\cdots \\&&&\vdots &\vdots &\vdots&\vdots&\vdots \end{array}$$ If you now rearrange the sum into column sums as suggested by the above arrangement, this yields $$e^{-1/e} = \frac1{e} + \sum_{n=1}^\infty \frac{n}{n!} -\frac12 \sum_{n=1}^\infty \frac{n^2}{n!}+\frac16 \sum_{n=1}^\infty \frac{n^3}{n!}-\cdots$$ or $$e^{-1/e} = \frac1{e} + \sum_{k=1}^\infty\left( \sum_{n=1}^\infty (-1)^{n+k} \frac1{k!} \sum_{n=1}^\infty \frac{n^k}{n!}\right)$$ Now for any given moderately small integer $$k$$, it is not too difficult (tried it by hand in each case) to use the same sort of rearrangement tricks to sum $$\sum_{n=1}^\infty \frac{n^k}{n!}$$ And the answers (starting at $$k=1$$, and including the proper sign and leading factorial) are $$\left\{\frac1{e}, 0, -\frac1{6e}, \frac1{24e}, \frac1{60e}, -\frac1{80e}, \frac1{560e} , \frac5{4032e}\right\}$$ You have to wonder where it is safe to stop: You would like a series which rapidly decreasing terms so that you can safely stop discarding a negative term, to give an expression $$E$$ where you know that $$e{-1/e} < E$$. The raw series shown does not lend confidence, but if you group terms in pairs you get $$\left\{\frac1{e}, 0, -\frac1{6e}, \frac1{240e}, \frac{61}{20160e}, -\frac{2257}{3628800e} \cdots \right\}$$ and that tells you you can confidently stop after the $$\frac{61}{20160e}$$ term, getting some number which is larger than $$e^{-1/e}$$.

When you do this you get $$e^{-1/e} < \frac{7589}{4032 e} \approx 0.69242$$ Finally, know $$\ln(2)$$ to four decimal places, and can do the comparison.

An interesting side relation the above "proves" is that $$e^{1-\frac1{e}} < \frac{7589}{4032}$$


# Closure of set of all differentiable functions in $(C[0,1],||.||_{\infty})$

Define $$D:=\{ f \in C[0,1] \mid f$$ is differentiable $$\}$$. Find $$\overline{D}$$ . Is $$D$$ open or closed ?

My attempt: I think $$D$$ is neither open nor closed in $$(C[0,1],||.||_{\infty})$$. consider the function $$f(x)=0$$ $$\,\forall\,$$ $$x \in [0,1]$$ , clearly $$f \in D$$. Let $$r$$ be any arbitrary positive real number. Then $$g \in B(f,r)$$ , where $$g(x) = r\mid x-\frac{1}{2}\mid$$. $$g$$ is a continuous function but it is not differentiable at $$x=\frac{1}{2}$$ . Hence $$B(f,r) \not\subseteq D$$ . This proves that $$D$$ is not an open set.

To show that $$D$$ is not closed, we want to construct a sequence of differentiable function which uniformly converge to a non-differentiable function. Consider the function $$f_n(x)$$ which is defined as below.

$$f_n(x)=\left \{ \begin{array}{ll} -x+\frac{1}{2}\left( 1-\frac{1}{n}\right) & \mbox{, if } 0\le x <-\frac{1}{n}+\frac{1}{2} \\ -\frac{n}{8}(2x-1)^2 & \mbox{, if } -\frac{1}{n}+\frac{1}{2}\le x <\frac{1}{n}+\frac{1}{2} \\ x-\frac{1}{2}\left( 1+\frac{1}{n}\right) & \mbox{, if } \frac{1}{n}+\frac{1}{2} \le x \le 1 \\ \end{array} \right.$$

The corresponding function to which it will converge is $$f(x)=\mid x-\frac{1}{2}\mid$$ . But $$f \not\in D$$, hence $$D$$ is not closed. This finishes the proof. Kindly verify if this proof is correct or not. I believe that $$\overline{D}$$ will be the whole metric space. But I am not able to prove this.

Definition 1: A point $$x \in X$$ is called a limit point of $$E\subseteq X$$ if $$B(x,r)\cap E \ne \emptyset$$ $$\,\forall\, r>0$$.

Definition 2(Interior of a set): Let $$S\subset X$$ be a subset of a metric space. We say that $$x \in S$$ is an interior point of $$S$$ if $$\,\exists\,$$ $$r > 0$$ such that $$B(x, r) \subset S$$ . The set of interior points of $$S$$ is denoted by $$S^o$$ and is called the interior of the set $$S$$

What will be $$D^o$$? is it the empty set?

• Indeed, $\overline D$ is all of $C([0, 1])$. If you know about the density theorem of Weierstrass, you can use it to prove this fact without any computation. Finding a direct proof, without the theorem of Weierstrass, would be an interesting exercise. Also, $D°=\varnothing$. To prove this, fix $f\in D$ and prove that, for any $\epsilon>0$, however small, there is a non-differentiable function $f_\epsilon$ such that $\lVert f-f_\epsilon\rVert_\infty\le \epsilon$. – Giuseppe Negro Jan 24 at 10:54
• I am sorry @GiuseppeNegro, I am not aware of the density theorem of Weierstrass. Can we give a prove without using this thm? – Sabhrant Jan 24 at 10:56
• Of course. Actually you have already done almost all. Here you approximated the function $x\mapsto |x-\tfrac12|$. Ok. Now you have to approximate a generic function $x\mapsto f(x)$. Try to do so by splitting $[0, 1]$ in sub-intervals of length $\frac1n$. – Giuseppe Negro Jan 24 at 11:09

To prove that the interior $$D°$$ is empty, you need to prove that for each $$f\in D$$ and for each $$\epsilon>0$$ there is $$f_\epsilon\notin D$$ such that $$\lVert f-f_\epsilon\rVert_\infty <\epsilon$$. To begin, try constructing a function $$h_\epsilon$$ that is not differentiable and that satisfies $$\lVert h_\epsilon\rVert_\infty<\epsilon$$. (Think of a tiny saw-tooth). Once you have this, you are finished: just let $$f_\epsilon:=f+h_\epsilon$$.
To prove that $$D$$ is dense, there are many ways, of course. I would do the following. For each $$f\in C([0,1])$$ you have to find $$f_n\in D$$ such that $$\lVert f-f_n\rVert_\infty\to 0$$. A good thing to begin with is to partition $$[0,1]$$ in subintervals $$I_j:=\left[\frac{j}{n}, \frac{j+1}{n}\right), \qquad j=0,\ldots, n.$$ Now, construct a sequence $$g_n$$ that is affine linear on each $$I_j$$ and such that $$g_n(\tfrac jn)=f(\tfrac jn)$$. You can easily prove that $$\lVert g_n-f\rVert_\infty\to 0$$. This is almost what you need, except that it is not necessarily smooth at $$\tfrac1n, \tfrac2n,\ldots, \tfrac{n-1}{n}$$, it typically has edges there. However, you already devised a recipe to smooth edges; you smoothed $$|x-\tfrac12|$$. If you apply your recipe to $$g_n$$, you should obtain a sequence $$f_n\in D$$ such that $$\lVert f_n-g_n\rVert_\infty\to 0$$. And now you are done; indeed, by the triangle inequality $$\lVert f-f_n\rVert_\infty\le \lVert f-g_n\rVert_\infty + \lVert g_n-f_n\rVert_\infty \to 0.$$
Yes, it is correct. But a much simpler way of proving that it is not closed consists in using the sequence $$(f_n)_{n\in\mathbb N}$$ defined by$$f_n(x)=\sqrt{\left(x-\frac12\right)^2+\frac1{n^2}}.$$Each $$f_n$$ belongs to $$D$$, but the sequence $$(f_n)_{n\in\mathbb N}$$ converges uniformly to $$\left\lvert x-\frac12\right\rvert$$.