How to solve a PDE where the boundary condition is an spatial ODE

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I have a PDE which i need to solve to get both the spatial and time related results. However i am confused as to how to pose the problem in matlab especially with the boundary conditions. I have attached the problem so that it is clear what i am talking about. The initial condition for lambda at t=0 is 0. The boundary conditions (at x = 0 and at x = L) is N_lambda = 0. Can anyone help me out here in defining the problem in matlab. Thank you very much in advance

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Bill Greene
Bill Greene 2023-4-20
I suggest you use the pdepe function to solve this PDE.
In PDE of the type you show, especially when the PDE represents a physical phenomenon, terms like your N_lambda are referred to as the "flux." pdepe makes it easy to define boundary conditions where the flux equals a specified value, e.g. zero.
pdepe requires you to create a function to define your PDE. For your example it would look something like this:
function [c,f,s] = pdeDefinition(x,t,lam,dlamdx)
epsi=1;
Vm=2;
DLam=3;
nd=4;
ip=5;
F=6;
c = epsi/Vm;
f = DLam/Vm*dlamdx - nd*ip/F;
s = S+r;
end
The boundary condition function, which you also have to write, would look something like this.
function [pl,ql,pr,qr] = bcFunc(xl,lam_l,xr,lam_r,t)
pl = 0;
ql = 1;
pr = 0;
qr = 1;
end
  6 个评论
Bill Greene
Bill Greene 2023-4-20
Regarding pdepe BC, this note I wrote several years ago might be useful.
Regarding the usage of pdepe in general, I strongly recommend starting with a simple PDE that you know the analytical solution to and experiment with that until you are comfortable with the syntax and options.
Sachin Hegde
Sachin Hegde 2023-4-20
Thank you very much for the notes. I was able to solve the PDE with your provided method. The PDE gives a solution in time and space. What i am now looking for is to use the last row of the results (at t = tend of the pdepe) to another PDE as initial condition. Since the result of the first PDE is a vector of size (1*n), how can it be passed into the inital condition of another pdepe solver. How to do it? When i try to pass an array in IC it is always a scalar zero.
1st PDE
Solution : (1*n) array
Use Solution as Initial condition for 2nd PDE

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