# Help with linear PDE problem

1. Sep 1, 2009

### tsetty2000

Here is the problem:
∂v(s,n)/∂n + ∂u(s,n)/∂s + ∂ξ(s,n)/∂s + An dc(s)/ds = 0 (1)
A1 ∂ξ(s,n)/∂n + ∂v(s,n)/∂s -c(s)+A2 v(s,n) + A3 c(s) = 0 (2)
∂u(s,n)/∂s + 2A2 u(s,n)=A2(ξ(s,n) + Anc(s)) -A1 ∂ξ(s,n)/∂s-A2nc(s) (3)

Unknowns: u(s,n),v(s,n),ξ(s,n)
Boundary conditions: v(s,1)=v(s,-1)= 0

I am trying to solve this set of PDEs by iteration and I am not sure if I am going about it the correct way. I have attached my attempt at a solution and it seems i am going in circles. Does anyone have a better idea? Someone suggested using Fourier analysis to solve the problem. I am reading that up now but i would really appreciate any ideas on how to start.

Thank you.

#### Attached Files:

• ###### attempt at solution.pdf
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2. Sep 10, 2009

### kosovtsov

Dear tsetty2000,

Your PDE system (if I do not misprint it) can be solved exactly by Maple 13 in the following way

sys:=[diff(v(s,n),n) + diff(u(s,n),s) +diff(xi(s,n),s) + A*n*diff(c(s),s) = 0,
A1*diff(xi(s,n),n) + diff(v(s,n),s) -c(s)+A2*v(s,n) + A3* c(s) = 0,
diff(u(s,n),s) + 2*A2* u(s,n)=A2*(xi(s,n) + A*n*c(s)) -A1*diff(xi(s,n),s)-A2*n*c(s)];

> ans:=pdsolve(sys,[xi,{u,v}]);

> pdetest(ans,sys);

The answer is bulky enough to be reproduced here (see att in Maple).

File size:
4.3 KB
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