Regards, Solution to PDE system by Maple 13

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In summary, the given conversation discusses a problem involving a set of three partial differential equations with unknowns u(s,n),v(s,n), and ξ(s,n) and boundary conditions for v(s,1) and v(s,-1). The individual equations and the suggested method of solving them are provided, with the idea of using Fourier analysis mentioned. However, it is also suggested to use Maple 13, which can solve the system exactly. The solution is provided in a separate attachment.
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tsetty2000
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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.
 

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  • attempt at solution.pdf
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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).
 

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  • tsetty2000_sys.mws
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FAQ: Regards, Solution to PDE system by Maple 13

What is a linear PDE?

A linear PDE (partial differential equation) is a type of mathematical equation that involves partial derivatives of a function. It is called "linear" because the unknown function and its derivatives appear only to the first power, and there are no products of the unknown function or its derivatives.

How do I solve a linear PDE?

The process for solving a linear PDE depends on the specific equation and its initial/boundary conditions. In general, the approach involves transforming the PDE into a simpler form (such as by using a change of variables), applying boundary/initial conditions, and then solving for the unknown function.

What are some common techniques for solving linear PDEs?

Some common techniques for solving linear PDEs include separation of variables, method of characteristics, and Fourier/Laplace transforms. Depending on the equation and its conditions, other methods such as finite difference or finite element methods may also be used.

Are there any software packages or tools available for solving linear PDEs?

Yes, there are many software packages and tools that can be used to solve linear PDEs. Some examples include MATLAB, Mathematica, Maple, and Python libraries such as SymPy and SciPy. These tools can help with both analytical and numerical solutions.

What are some real-world applications of linear PDEs?

Linear PDEs have many practical applications in fields such as physics, engineering, and finance. They are commonly used to model phenomena such as heat transfer, diffusion, fluid flow, and electromagnetic fields. They are also used in image processing and signal analysis.

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