Similarity solutions of partial differential equations using DESOLV☆
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, , admits the five symmetry vector fields: , , , , and . These can be readily obtained using pre-existing functions in DESOLV. We shall focus on the symmetry , 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 , so that 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.
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This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).