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Problem

Approximate the solution of a Poisson equation

u_xx(x, y) + u_xx(x, y) = g

on a rectangular region R = [x_a, x_b] × [y_a, y_b] with Dirichlet boundary conditions given by a function ∂u(x, y).

Assumptions

The function ∂u(x, y) must be piecewise continuous. We will also assume that x_b − x_a = hn and y_b − y_a = hm for some integer values of n and m.

Tools

We will use the centred divided-difference formula for the partial derivatives and linear algebra.

Process

Divide the interval [x_a, x_b] into n sub-intervals by setting x_i = x_a + ih for i = 0, 1, 2, ..., n and y_i = y_a + jh for j = 0, 1, 2, ..., m. Let u_{i, j} represent the approximation of the solution u(x_i, y_j).

At each interior point (x_i, y_j) where i = 1, 2, ..., n − 1 and j = 1, 2, ..., m − 1, evaluate the finite-difference equation:

u_{i + 1, j} + u_{i − 1, j} + u_{i, j + 1} + u_{i, j − 1} − 4 u_{i, j} = gh²

This defines a system of (n − 1)(m − 1) linear equations and (n − 1)(m − 1) unknowns. We can now solve these for the approximations u_i, j.

Topic 15.2: Elliptic Partial-Differential Equations (HOWTO)

Problem

Assumptions

Tools

Process