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Topic: Solving Partial Differential Equations with Mathematica?
Replies: 2   Last Post: Nov 25, 2012 6:57 PM

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Nasser Abbasi

Posts: 5,674
Registered: 2/7/05
Re: Solving Partial Differential Equations with Mathematica?
Posted: Nov 24, 2012 8:34 PM
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On 11/24/2012 10:53 AM, Vladimir Bondarenko wrote:
> Hello all,
>
> What could be a couple of best books on solving PDEs using Mathematica?
>
> Much thanks in advance!
>


Hello Vladimir,

It is not clear if you asking on using Mathematica's NDSolve
to solve PDE's or on just a general use of Mathematica to
solve PDE's writing your own implementation.

This is like with Matlab, where there is a PDE toolbox, or
one can write their own implementation of a solver from scratch.

If you are asking on using Mathematica NDSolve, then I would
not bother with a book, since all the information is online:

1) Advanced Numerical Differential Equation Solving in Mathematica
http://www.wolfram.com/learningcenter/tutorialcollection/AdvancedNumericalDifferentialEquationSolvingInMathematica/

2)help
http://reference.wolfram.com/mathematica/ref/NDSolve.html
http://reference.wolfram.com/mathematica/howto/SolveAPartialDifferentialEquation.html

3)I wrote this function to obtain all options for a command
in a table format. You can use it to find all the options for
NDSolve.

--------------------------------
getList[name_String] := Module[{options, idx},
options = Names[name <> "`*"];
options = ToExpression /@ options;
options = {#, Options[#]} & /@ options;
idx = Range[Length[options]];
options = {#[[1]], TableForm[#[[2]]]} & /@ options;
options = Insert[options[[#]], #, 1] & /@ idx;
options =
Insert[options, {"#", "Option", "Options to this option"}, 1];
Grid[options, Frame -> All,
Alignment -> Left,FrameStyle ->
Directive[Thickness[.005], Gray]]
];
-------------------------------

to use it

getList["NDSolve"]

You can see the huge amount of options and different methods
supported there.

4)Based on what I read on the net, version 9 might have
Finite elements solver added. I do not know for sure ofcourse,
but that is the rumor.

5)I wrote few demos to solve some PDE's in Mathematica
(Using known standard finite difference schemes I learned
in a PDE class, not using NDSovle):

http://demonstrations.wolfram.com/SolvingThe2DPoissonPDEByEightDifferentMethods/
http://demonstrations.wolfram.com/SolvingThe2DHelmholtzPartialDifferentialEquationUsingFiniteD/
http://demonstrations.wolfram.com/SolvingTheDiffusionAdvectionReactionEquationIn1DUsingFiniteD/

6)for books, I see on amazon some like this
http://www.amazon.com/Partial-Differential-Equations-Mathematica-Vvedensky/dp/0201544091
which is available for free download (the notebooks) here
http://library.wolfram.com/infocenter/Books/3232/

but I did not use or have it so can't comment. There are more on
amazon.

But I do not use books specific to Mathematica. I have general books
on PDE's numerical solutions, which I can implement in
Mathematica. I prefer to implement something myself than
use a blackbox solver (unless I am just doing something quick
to look at), since I learn more that way.

Instead of spending the time of learning how to use a blackbox
solver with all its options and suboptions, I can spend the time
learning the actual algorithm used to solve a PDE, which is
more useful for me.

Even with Matlab, I do not use the pde toolbox, but
wrote my own code. (But I am a student, so things might
be different for others)

--Nasser



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