How to write 2D heat conduction exact solution for Inf n?

Hi, I'm Currently going to perform numberical analysis with a Exact solution of the 2-D heat conduction equation.

That, H is y-grid length, L is x-grid Length, T1 is bottom Temperature. I try to write code like this.

However, the figure in the code is wrong and the figure I want is as follows.

how can i derive that equations? Thank you for your reply.

clc;
clear;
close all;
%% grid input
xnum = 51; 
ynum = 101;

%% Given Data
xmax = 1;  % L
ymax = 2;  % H
T1 = 100;  % bottom Temperature
T2 = 0; 
T3 = 0;
T4 = 0; 

%% Data
dx = xmax/(xnum-1); 
dy = ymax/(ynum-1); 
beta = dx/dy;
alpha = -2*(1+beta^2);
k = 1; 

%% Boundary Condition 
T = zeros(xnum,ynum,k); % Initial Condition Temperature = 0 
for i = 1:1:xnum
 T(i,1,1) = T1;
 T(i,ynum,1) = T3;
end
for j = 2:1:ynum-1
 T(1,j,1) = T2;
 T(xnum,j,1) = T4;

end
% Calcultae T = T_1 * [2*sum( f(sinh) )]
for j = 2 : 1: ynum-1
    for i = 1 : 1 : xnum
        for n = 1 : 1 : 100
        T(i,j) = T(i,j) + 2*T1*(1 -(-1)^n)/(n*pi) * sinh(n*pi*(ymax-j*dy)/1) / sinh(n*pi*ymax/1) * sinh(n*pi*i*dx);
        end
    end
end

figure(1)
contourf(T(:,:,1));
% contourf(T(:,:,1),'ShowText','on');
hold on;
view(90,270);
grid on
hold on
axis('equal')
xticks([1 0.25*(ynum-1)+1 0.5*(ynum-1)+1 0.75*(ynum-1)+1 ynum])
xticklabels({'0','0.5','1','1.5','2'})
yticks([1 0.5*(xnum-1)+1 xnum])
yticklabels({'0','0.5','1'})
xlabel('y  [ ft ]')
ylabel('x  [ ft ]')
colormap jet
colorbar

[Torsten]

I doubt your formula to compute T is correct.

Answer 1

[Alan Stevens]

Your term sinh(n*pi*x/L) should be sin(n*pi*x/L). See the following code (which can be made more efficient!):

L = 1;  H = 2;
nx = 51;  ny = 101;
dx = L/(nx-1);  dy = H/(ny-1);
x = 0:dx:L;  y = 0:dy:H;
T1 = 100;

m = 31;  

T = zeros(numel(y), numel(x));

 for j = 1:numel(y)
     for k = 1:numel(x)
         
        for n = 1:m
            u = (1 - (-1)^n)/(n*pi);
            sinhterms = sinh(n*pi*(H-y(j))/L)/sinh(n*pi*H/L);
            sinterm = sin(n*pi*x(k)/L);
            T(j,k) = u*sinhterms*sinterm + T(j,k);       
        end
         
    end 
end 
T = 2*T1*T;

figure
contourf(x,y,T,9)
axis equal
colormap jet
colorbar

[병권 이]

Oh, thank you! I couldn't see the misspelling of the exact sol. and kept struggling in vain.