# Steady state heat equation:-((∂^2 T)/(∂x^2 )+(∂^2 T)/(∂y^2 )=-Q(x,y)

1. Aug 5, 2014

### range.rover

Hi friends i am trying to solve heat equation for a single time domain,with Finite Difference Method. I wrote a program in MATLAB,it sucessfully worked, but as i increse the number of points, my program is taking lots of time to give out result.

Can anybody help me in increasing the speed of my Program:-

clear;
close all;
clc;
n = 5;% n grids and has n - 2 interior points per dimension

x = linspace(0,1,n);% line in x direction of 0 to 1 and divided into n parts

dx = x(2)-x(1); % distance between grids

y = x;

dy = dx;

k=1; %thermal conductivity

Q=300 ; % heat source
m=(n-2)^2;

%Right hand side matrix

B=zeros(m,1);

if (mod(m,2)==0)
B(m/2,1)=Q;
else
B((m+1)/2,1)=Q;
end

%Main Matrix

Tsol = zeros(n);
%boundry conditions
Tsol(1,1:n) = 0; %TOP
Tsol(n,1:n) = 0; %BOTTOM
Tsol(1:n,1) = 0; %LEFT
Tsol(1:n,n) = 0; %RIGHT

tic
error = 1; z = 0;
while error > 0.000001
z = z+1;
c=1;
Torg = Tsol;
for i = 2:n-1
for j = 2:n-1
Tsol(i,j) =abs(((dx^2*B(c,1))+((Torg(i+1,j)+Torg(i-1,j)+Torg(i,j+1)+Torg(i,j-1))*k))/(4*k));
c=c+1;
end
end

error = max(max(abs(Torg-Tsol)));
end

toc

%% plotting

figure, contour(x,y,Tsol),

2. Aug 13, 2014