Paper (PDF)
Abstract
This work presents a general discussion of and plan for a computational
approach towards the analytical solving of Partial Differential Equations
(PDEs), as well as a symbolic computing implementation of the first part of
this plan as the PDEtools software-package of commands. The package consists of
a PDE-solver (called pdsolve) and some other tool-commands for working
with PDEs. A brief overview of the most relevant commands of the package is as
follows:
-
- pdsolve looks for the general solution or the complete separation
of the variables in a given PDE. There are no "a priori" restrictions as to
the kind of PDE that the program can try to solve. This command uses, as often
as possible standard methods, when the PDE matches the corresponding pattern,
and a heuristic algorithm for separating the variables otherwise. Partial
separation of variables is handled by automatically reentering the program
with a smaller problem, resulting in a wide combination of standard methods
and separation of variables in the solution of a single PDE. Furthermore,
the User can optionally participate in the solving process by giving the
solver an extra argument (the HINT option) as to the general functional form
of the indeterminate function.
- dchange performs changes of variables in PDEs and other algebraic objects
(integro-differential equations, limits, multiple integrals, etc...). This
command is useful to change the format of a PDE from one that is difficult to
solve to one that is solvable by the system. In addition, dchange can be used
to analyze underlying invariance groups of a PDE, since it works with
changes of both the independent and the dependent variables, automatically
extending the transformations to any required differential order.
- sdsolve looks for a complete (or partial) solution (or uncoupling) of a
coupled system of ODEs (linear or not).
- mapde maps PDEs into other PDEs, more convenient in some cases, within the
frame of the possibilities of the system. The mappings tested and implemented
up to now are:
-
- Linear 2nd order PDEs, with two differentiation variables, into PDEs in
canonical form (two different types). Though the success depends
strongly on the proposed problem, in principle this program works both
with PDEs with constant or variable coefficients.
- PDEs that explicitly depend on the indeterminate function into PDEs
which does not (i.e. only depends through derivatives).
- strip evaluates the characteristic strip associated to a first-order PDE.
- pdtest tests a solution found by pdsolve for a given PDE by making a
careful simplification of the PDE with respect to this solution.
The goals of this work were to propose a computational strategy for the
problem and to present a first concrete implementation of a PDE-solver in
general purpose symbolic computing systems. Routines for the analytical
solving of systems of PDEs are now under development and shall be reported
elsewhere.
Submitter's Name: Edgardo S. Cheb-Terrab.
Submitter's Institution: Instituto de Fisica,
Universidade do Estado do Rio de Janeiro.
Address of Institution: Rua S"ao Francisco Xavier 524,
Rio de Janeiro RJ 20550, Brasil.
Submitter's EMAIL address: terrab@bruerj.bitnet
Submitter's Telephone number: 55-21-239-6981
Authors of the Paper: E.S.Cheb-Terrab and K. von B"ulow,
from the Submitter's Institution.