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//Eg-15.2
//pg-608
clear
clc
// Using D in the place of greek alphabet delta
// Taking square grid => Dx = Dy = 0.25
// Applying central difference approximation to the second derivatives, we obtain
// [T(i+1,j) - 2*T(i,j) + T(i-1,j)]/(Dx)^2 + [T(i,j+1) - 2*T(i,j) + T(i,j-1)]/(Dy)^2 = 0. i = 1,2,3; j = 1,2,3.....this can be simplified as
// T(i+1,j) + T(i-1,j) + T(i,j+1) + T(i,j-1) - 4*T(i,j) = 0
//Applying the above equation to the 9 points analytically leaves us with 9 equations and 9 variables T11 to T33. This can be written in the matrix equation form Ax = B.
A = [4 -1 0 -1 0 0 0 0 0;-1 4 -1 0 -1 0 0 0 0;0 -1 4 0 0 -1 0 0 0;-1 0 0 4 -1 0 -1 0 0;0 -1 0 -1 4 -1 0 -1 0;0 0 -1 0 -1 4 0 0 -1;0 0 0 -1 0 0 4 -1 0;0 0 0 0 -1 0 -1 4 -1;0 0 0 0 0 -1 0 -1 4];
B = [65;25;125;40;0;100;90;50;150];
printf('Solving the Equation Ax = B will give the values of Temperatures, where A = \n')
disp(A)
printf('\nand B = ')
disp(B)
printf('\nTherefore the matrix representing T11 to T33 is \n')
x = inv(A)*B;
disp(x)
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