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Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to

a) partial differential equations, b) stochastic differential equations?

If yes, what experiences do you have with these?

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up vote 3 down vote accepted

I have used Wolfram Mathematica extensively in my undergraduate course so far. Although the PDEs and systems of PDEs I have encountered have not been overly complicated, Mathematica is able to solve them in closed form most of the time. While not a "specialised" CAS for PDEs/SDEs, it gave me the closed form solutions I was looking for.

This link may be useful in terms of gauging what you can do.

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Most ODEs, nevermind PDEs and SDEs don't have what one would usually call "closed form" solutions.

Are you interested in a special class of equations (e.g. linear, constant coefficient, on flat manifolds without boundary) for which closed form solutions are likely to exist?

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The background of my question is a financial one: so in most cases parabolic pdes (mathematica is not able to solve these!) and sdes with wiener processes and jumps. – vonjd Oct 30 '09 at 12:47

For PDEs, the PDEtools package in Maple might be of use. I don't know exactly which types of PDEs it can solve, but the author and maintainer of this package, Edgardo Cheb-Terrab, has a background in theoretical physics, so I suspect that is its main emphasis.

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For usual (non-stochastic) PDEs indeed consider giving Maple a try. You don't even have to bother too much with the PDEtools package, just try Maple's command pdsolve and see for yourself.

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