Derive the linear system for the finite difference approxima

Derive the linear system for the finite difference approximation of the elliptic equation 2T/x^2/y^2 = f(x, y). The domain is in the unit square, Deltax = Deltay = 0.01, and the boundary conditions are given by T(x,0) = 1 - x, T(1,y) = y, T(0,y) = 1, T(x, 1) = 1. 6.6 For Exercise 6.5, if f(x, y) = -pi^2 sin(pi x) sin(pi y), then the analytic solution to the elliptic equation 2T/x^2/y^2 = f(x, y), with the same boundary conditions as in Exercise 6.5, is given by T(x, y) = 1 - x + xy + (1/2) sin(pi x) sin(pi y) (a) Use the finite difference scheme of Section 6.3.4 to approximate the values of T at the interior points with Deltax = Deltay = 1/n,n = 4,8,16. (b) Compare the values obtained in (a) with the exact solution.

Solution

The above problem is Poisson Equation of Diritchlet type . It can be expressed in form of discrete equations .

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Finally according to new nomenaclature , any node T(x,y) is now denoted by T(i,j) giving the equations as

T(i+1,j) + T(i-1,j) +T(i,j+1) + T(i,j-1) - 4*T(i,j) = h*h*f(i,j)