Similarity solutions of partial differential equations using DESOLV

https://doi.org/10.1016/j.cpc.2007.03.005Get rights and content

Abstract

We present and describe new reduction routines included in DESOLV which, in many cases, may allow the complete automation of the determination of similarity solutions of partial differential equations.

Program summary

Title of program: DESOLV

Catalogue identifier: ADYZ_v1_0

Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYZ_v1_0

Program obtainable from authors: [email protected], [email protected] and from CPC Program Library, Queen's University of Belfast, N. Ireland

Computer for which the program is designated and others in which it has been tested: Platforms supported by the Maple (version 9 or higher) computer algebra system

Operating systems under which the program has been tested: Linux, Windows XP

Programming language used: Maple internal language

Memory required to execute typical problem: Dependent on problem (small ≈ MB, large ≈ GB)

Number of bits in a word: Dependent on Maple distribution (supports 32 and 64 bit platforms)

Number of processes used: 1

No. of lines in distributed program, including test data, etc.: 6185

No. of bytes in distributed program, including test data, etc.: 59 422

Distributed format: tar.gz

Nature of the physical problem and method of solution: Systems of differential equations occur often in many theoretical and applied areas. In many cases, exact solutions are required as numerical methods are not appropriate or applicable. Indeed, exact solutions of systems of partial differential equations arising in fluid dynamics, continuum mechanics and general relativity are of considerable value for the light they shed into extreme cases which are not susceptible to numerical treatments. One important source of exact solutions to differential equations is the application of the group theoretic method of Lie. Such solutions found by Lie's method, are called invariant solutions. Essential to this approach is the need to solve overdetermined systems of “determining equations”, which consist of coupled, linear, homogeneous, partial differential equations. Typically, such systems vary between ten to several hundred equations. Clearly in the case of sets of equations consisting of about 100 equations or more, the prospect of finding solutions to such systems with just pencil and paper would certainly be quite daunting. DESOLV, which runs under Maple, attempts to automate as much as possible the process of determining these invariant solutions. The program has modular structure and not only uses basic features of Maple but has independently built-in routines to augment or assist

Typical running time: Dependent on problem (small ≈ second, large ≈ hours)

Restrictions on the complexity of the problem: Sufficient amount of memory and the nature of the determining system of equations

Introduction

In a recent article [1], we announced and discussed the package DESOLV, written for Maple, for the determination of Lie symmetries of differential equations. In the initial version of our software, our aim was to produce as capable a package as possible with many sophisticated tools which would facilitate the integration of the determining equations for the similarity groups. Indeed some systems of differential equations tested (e.g., the Vaidya system [2], [3]) required, in order to avoid “failure”, very specific settings of certain parameters in the various routines (thereby calling other “assist” routines) for the complete determination of the symmetries. In producing our package, we were also mindful of the requirement of user-friendliness and to automate as much of the process as possible. Some time ago, a comprehensive study [4] was carried out where DESOLV was compared to earlier versions of MATHLIE [5], BIGLIE/LIE [6], [7], DIMSYM [8], and CRACK/LIEPDE [10], [11]. It was shown that DESOLV performed exceedingly well, compared to these packages, when applied to a large testbed of differential equations. Also, it was easy (conventional Maple input) to use and most resembled the standard mathematical notation. Hence, we believe we have achieved our original goals with the initial version of DESOLV.

Of course, as in many applied cases, one is frequently interested in generating some explicit (invariant) solutions to partial differential equations (PDEs) and then hopefully there will be a solution which will satisfy the required initial/boundary conditions. As a first step in obtaining these solutions, canonical or similarity coordinates need to be found by integrating the group characteristic equations corresponding to a symmetry. Then, one executes a change of variables on the PDEs and as a result the number of independent variables is reduced by one. Finally, one hopes that these “reduced” PDEs will be easier to integrate.

In this article, we present and describe new functions in DESOLV which automate the reduction of variables of a system of PDEs given that it admits a Lie symmetry [12]. We then show how, in many cases, this leads directly to invariant solutions with very little to no user intervention.

Section snippets

An illustrative example: Burgers' equation

It is well known that Burgers' equation, uxx+uux+ut=0, admits the five symmetry vector fields: x, tx+u, t, xx+2ttuu, and xtx+t2t+(xtu)u. These can be readily obtained using pre-existing functions in DESOLV. We shall focus on the symmetry xtx+t2t+(xtu)u, and first summarize the required basic steps to find the corresponding invariant solution. To begin with, the coefficient of the x-direction depends on x and t whereas the coefficient of the t-direction depends only on t. Thus by

An algorithm for symmetry reduction

To begin with, suppose a system Ω has k PDEs and n variables, with m independent variables and l dependent variables (n=m+l), so thatΩ={Ω1(x),Ω2(x),,Ωk(x)},x=(x1,x2,,xm,xm+1,,xn). Our symmetry reduction process can be divided into four main steps:

  • 1.

    Find a “parametric” variable based on a given Lie symmetry.

  • 2.

    Solve the system of characteristic equations derived from the Lie symmetry to find the transformation relating other variables to a particular one (parameter).

  • 3.

    Apply the transformation to the

Applications

In this section, we will determine invariant solutions of four more interesting examples. The aim is to demonstrate how the routines in DESOLV can be successfully used on their own and in combination with existing routines in Maple.

Discussion and future development

The Maple routines within DESOLV should prove quite useful to the large community of Maple users who wish to investigate invariant solutions of differential equations. Specifically, in many “simpler” cases, the determination of the similarity groups and the subsequent reduction and integration of partial differential equations can be almost fully automated with, essentially, just the use of liesolve and (possibly) dsolve or pdsolve. In addition, we have also demonstrated that the routines in

Acknowledgements

One of us (K.T. Vu) would like to thank CardSmart Consultant for financial support.

References (25)

  • T. Wolf, The Computer Algebra Package CRACK for investigating PDEs, with A. Brand, in: Proceedings of ERCIM, Partial...
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