Hello all,
I have a set of coupled PDE with well defined domain / boundary conditions, but it's beyond my knowledge to solve it. Could anyone suggest a good method to analytically or numerically solve it? Thanks in advance!
\left[ x \partial_x + y \partial_y + H(x)+ H(y) \right] T_0(x,y) ={}& 1+H(x) T_y(x,y) +H(y) T_x(x,y), \\
\left( 1- \partial_x \right )T_x ={}& T_0,\\
\left( 1- \partial_y \right )T_y ={}& T_0.\\
\\
\\
\text{Domain } \Omega = \left\{ (x,y) \vert x>x_0, y
You can use the Fourier transform method or the Laplace transform method
To solve it numerically, I would suggest Finite Element Method
or Pseudo-spectral method.
Dear Peyman: Thanks for the suggestion. I was also thinking about FEM but I'm not entirely sure how to implement the method to 1st-order and coupled system. Do you have good suggestion / reference for me? Thanks in advance!
Dear M.A. Zacky: will check the method. On top of my head, does this work with first-order system? Thanks!
There are very many books on Finite Element method and its implementation such as
"Understanding and Implementing the Finite Element Method by Mark S. Gockenbach ".
For using pseudo-spectral (spectral collocation) method you can use my paper "PRECONDITIONED SPECTRAL COLLOCATION METHOD
ON CURVED ELEMENT DOMAINS USING THE
GORDON-HALL TRANSFORMATION".
best
Peyman
The equations seem to constitute a system of linear PDE. Therefore (after writing the system in variational formulation) it is not difficult to use a standard FEM-oriented language, like FreeFem++, http://www.freefem.org/ff++/, or FEniCS, http://fenicsproject.org/ to obtain a approximations in different domains. In the documentation of those packages you can find example for some coupled PDE systems.
About instabilities near $x=y$, I think that a numerical analysis should be made if your FEM tests confirm what finite differences suggest.
Thanks, Peyman. This book seems to be a perfect introduction to me to pick up what I left (I learned FEM from a rushy numerical analysis / physics course and never had a chance to implement it since turning my term project).
J. Rafael Rodriguez Galvan: thanks for the input! Will check the software's!
you can also try COMSOL...It is very good in solving algebraic and Partial differential eqns
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