Question

Find the finite-difference solution of the heat-conduction problem
PDE: ut = uxx 0 < x < 1, 0 < t < 1
BCs:
⇢
u(0, t) = 0
ux(1, t) = 0
0 < t < 1
IC: u(x, 0) = sin(pi x) 0 x  1
for t = 0.005, 0.010, 0.015 by the explicit method. Assume

Answer 1

solution:

function [y]=pdeeq()

format long;
h=0.1;
k=0.005;
l=k/(h^2);

x=0:h:1;
t=0:k:0.015;

for i=1:length(x)
u(i,1)=sin(pi*x(i));
end
for i=1:length(t)
u(1,i)=0;
u(length(x),i)=0;
end

for j=1:length(t)-1
for i=2:length(x)-1
u(i,j+1)=l*(u(i+1,j)+u(i-1,j))+(1-2*l)*u(i,j);
end
end

disp(u);

Solution:

    

             t=0                             t=0.005                  t=0.010                t=0.015

                0                   0                   0                   0
   0.309016994374947   0.293892626146237   0.279508497187474   0.265828377610012
   0.587785252292473   0.559016994374947   0.531656755220025   0.505635621484342
   0.809016994374947   0.769420884293813   0.731762745781211   0.695947727757254
   0.951056516295154   0.904508497187474   0.860238700294483   0.818135621484342
   1.000000000000000   0.951056516295154   0.904508497187474   0.860238700294484
   0.951056516295154   0.904508497187474   0.860238700294484   0.818135621484342
   0.809016994374947   0.769420884293814   0.731762745781211   0.695947727757254
   0.587785252292473   0.559016994374947   0.531656755220025   0.505635621484342
   0.309016994374948   0.293892626146237   0.279508497187474   0.265828377610013
                   0                   0                   0                   0