MATLAB: Solving one-dimensional PDE’s using the PDE Toolbox

heat equationone dimensionpde toolbox

Hi,
I've been trying to solve a non-linear, heat-equation-type system of PDE's using the 'pdepe' function, with only one dimension in space. However, for many sets of parameter values, the solver exhibits unstable behaviour (oscillations, etc). I think that my problem demands a more sophisticated solver, due to nonlinearities and discontinuities. Hence, I am now trying to use the PDE Toolbox, hoping that it would be able to handle my problem, since it has an adaptive mesh algorithm, etc. However, I was unable to figure out how to use the PDE Toolbox for one-dimensional problems, nor was I able to find any examples of this.
Is it possible to use the PDE Toolbox to solve one-dimensional problems? I'm find with using the command-line interface (I don't necessarily need to use the GUI). If this is possible, I'd greatly appreciate it if someone could provide me with some code that solves a simple heat equation PDE (one dimension in space) using the PDE Toolbox, just to get me on the right foot (since, for some reason, I get really lost when I try to read the documentation for the PDE Toolbox).
Thanks,
Abed

Best Answer

Hi,
Solving a 1-D PDE with PDE Toolbox is fairly straightforward. You just define a rectangular region of the appropriate width and arbitrary height. On the two edges at y=constant, you want a zero-Neumann BC. On the edges at x=constant, the BCs should not vary in y.
I've appended a very simple example of time-dependent heat transfer in a bar below.
Based on your description of the problem you are trying to solve, however, I can't think of any reason why PDE Toolbox should give a better solution than pdepe.
Bill
function simple1DTest
% 1D transient heat transfer in x-direction
h =.1; w=1; % width equal one, height is arbitrary (set to .1)
g = decsg([3 4 0 w w 0 0 0 h h]', 'R1', ('R1')');
[p, e, t]=initmesh(g, 'Hmax', .05);
b=@boundFile;
c=1; a=0; d=1;
f='100*x'; % heat load varies along the bar
u0 = 0; % initial temperature equals zero
tlist = 0:.02:1;
u=parabolic(u0, tlist, b,p,e,t,c,a,f,d);
figure; pdeplot(p,e,t, 'xydata', u(:,end), 'contour', 'on'); axis equal;
title(sprintf('Temperature Distribution at time=%g seconds', tlist(end)));
figure; plot(tlist, u(2,:)); xlabel 'Time'; ylabel 'Temperature'; grid;
title 'Temperature at tip as a function of time'
end
function [ q, g, h, r ] = boundFile( p, e, u, time )
N = 1; ne = size(e,2);
q = zeros(N^2, ne); g = zeros(N, ne);
h = zeros(N^2, 2*ne); r = zeros(N, 2*ne);
% zero Neumann BCs (insulated) on edges 1 and 3 at y=0 and y=h
% and the right edge, edge 2
for i=1:ne
  switch(e(5,i))
    case 4
      h(1,i) = 1; h(1,i+ne) = 1;
      r(i) = 500; r(i+ne) = 500; % 500 on left edge
  end
end
end
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