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MathGroup Archive 1998

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Re: PDEs & Mathematica.


  • To: mathgroup@smc.vnet.net
  • Subject: [mg10775] Re: PDEs & Mathematica.
  • From: Lars Hohmuth <larsh@wolfram.com>
  • Date: Thu, 5 Feb 1998 00:58:22 -0500
  • Organization: Wolfram Research, Inc.
  • References: <199801270810.DAA01319@smc.vnet.net.> <6atasu$jlr$14@dragonfly.wolfram.com>

seanross@worldnet.att.net wrote:
>
> William Golz wrote:
> >
> > The "online book" does not describe how to plug in initial and boundary
> > conditions for a PDE; and when I try to do it ODE style, which is
> > explained I get error messages. If anyone knows the proper syntax for
> > solving PDEs with boundary conditions, I would appreciate some advice.
> >
> > I would also appreciate any information on good books dealing with PDEs
> > and Mathematica.
> >
> > ____________
> > William Golz
> > Department of Civil & Environmental Engineering Louisiana State
> > University
> > Baton Rouge, Louisiana 70803
> >
> > E-mail: wgolz@unix1.sncc.lsu.edu
> > Phone:  (318)237-8353
>
> The DSolve and NDSolve do not contain routines for partial differential
> equations.  I am not aware of any other built-in functions that have
> routines for partial differential equations. --
> Remove the _nospam_ in the return address to respond.

Actually, both DSolve and NDSolve have routines for handling certain
classes of partial differential equations. More specifically, DSolve
uses separation of variables and symmetry reduction, while NDSolve uses
the method of lines for 1+1 dimensional PDEs.

You usually specify initial conditions exactly like in the ODE case, but
keep in mind that solving PDEs is a much harder problem than ODEs. For
example, partial differential equation may not have a general solution.
SO it would be helpful to know exactly which equations you are trying
to solve.

Here is an example from the online documentation. It finds a numerical
solution to the wave equation with the initial condition
y[x,0]řxp[-x^2]. The result is a two≠dimensional interpolation
function.

In[1]:NDSolve[{D[y[x, t], t, t] –D[y[x, t], x, x],
    y[x, 0] –Exp[-x^2], Derivative[0,1][y][x, 0] –0,
       y[-5, t] –y[5, t]}, y, {x, -5, 5}, {t, 0, 5}] Out[1]{{y\[Rule]InterpolatingFunction[{{-5,5.},{0.,5.}},"<>"]}}

If general solutions don't exist, the standard package
Calculus`DSolveIntegrals` can be used to find complete integrals of the
PDE. Additionally, there are a couple of packages for calculating Lie
and Lie-Backlund symmetries available from www.mathsource.com.

There are a number of books about solving differential equations with
Mathematica, take a look at
http://store.wolfram.com/catalog/books/de.html .


Some more information is available in sections 3.5.10 and 3.9.7 of the
Mathematica Book.


Lars Hohmuth
Wolfram Research, Inc.



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