# Solving second order PDE

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Lotuny Lee on 26 Jun 2020
Edited: Lotuny Lee on 27 Jun 2020
Hi, I am trying to solve the following pde with initial condition CA(0,r)=0 and boundary conditions CA(t,0)=F(t) and CA(t,5)=0.
, where D_A and gamma_A are known constants.
I tried using pdepe but was told that left boundary condition would be ignored when m=1 (cylindrical symmetry).
Then I tried discretizing space variable r before using ode15s, but was confused about how to construct the equation exactly.
Can anybody help?
##### 2 CommentsShowHide 1 older comment
Lotuny Lee on 27 Jun 2020
Edited: Lotuny Lee on 27 Jun 2020
@darova Hi, it is a reaction-diffusion equation. And I believe the middle part should be .

Bill Greene on 27 Jun 2020
Edited: Bill Greene on 27 Jun 2020
The reason that pdepe imposes a boundary condition of the flux equal zero at the
center is that this is required for the problem to be mathematically well-posed.
Imposing a prescribed temperature at the center would require that the flux go to
infinity there.
An easy way to understand this is to solve the problem with the left end a small distance
from the center and with a fine mesh. I have attached a short script below that shows this.
r0=1e-6;
x = linspace(r0,1,1000);
tf=1;
t = linspace(0,tf,40);
pdeFunc = @(x,t,u,DuDx) heatpde(x,t,u,DuDx);
icFunc = @(x) heatic(x);
bcFunc = @(xl,ul,xr,ur,t) heatbcDirichlet(xl,ul,xr,ur,t);
m=1;
sol = pdepe(m, pdeFunc,icFunc,bcFunc,x,t);
figure; plot(t, sol(:,end)); grid on; title 'Temperature at outer surface'
figure; plot(t, sol(:,1)); grid on; title 'Temperature at center'
figure; plot(x, sol(end,:)); grid; title 'Temperature at final time'
end
function [c,f,s] = heatpde(x,t,u,DuDx)
c = 1;
f = DuDx;
s = 0;
end
function u0 = heatic(x)
u0 = 0;
end
function [pl,ql,pr,qr] = heatbcDirichlet(xl,ul,xr,ur,t)
pl = ul-1;
ql = 0;
pr = 0;
qr = 1;
end
Lotuny Lee on 27 Jun 2020
Thanks a lot! This looks like a feasible alternative.

J. Alex Lee on 27 Jun 2020
I believe that pdepe is available with base matlab.
It appeas to be able to do the space discretization automatically for you if you
Lotuny Lee on 27 Jun 2020
Thanks for answering, but my issue with pdepe is that my boundary condition would be ignored.

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