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1 -s . Session 1220

Visualization of PDE Solutions Using Implicit Methods and MATLAB

Raymond G. Jacquot, Jerry C. Hamann Electrical Engineering Department University of Wyoming

Abstract In many engineering curricula the formal techniques of the solution of partial differential equations are not studied, however, all such curricula examine problems described by such equations. An alternative to studying the formal solutions is to examine the solutions of such problems by numerical methods. The authors have found MATLAB particularly attractive for the solution of such problems because of the ease with which linear algebra problems are solved and displayed in that computing environment.

Introduction Physical systems governed by partial differential equations have become a part of most undergraduate engineering curricula although many of these curricula do not require formal mathematical training in such topics. Typical subject matter with such content includes vibrations, acoustics, transport phenomena and electromagnetic theory. Students without formal mathematical training in partial differential equations are almost helpless to understand the solution to such a problems in their engineering courses. An alternative to teaching analytical techniques of mathematical physics is to explore the solution to such problems in an approximate manner. Two ways to approach these approximate solutions are via numerical analysis or by the weighted residual approaches of Galerkin and Ritz. The former method employing MATLAB is the topic of this paper while the latter will be explored in a subsequent paper. For three decades the numerical solution to partial differential has been explored by practicing scientists, engineers and graduate students in engineering, and a decade ago it was still a formidable task often employing a library of general purpose subroutines to implement a particular numerical solution technique. The recent availability of MATLAB for desktop computing has provided a computing environment with robust built-in routines for matrix manipulation and seamless two and three dimensional graphics for presentation of the solutions thus obtained. In short, using MATLAB turns efforts the duration of which was formerly measured in days to durations of a few hours. In the past, implicit methods were often avoided because of the need to solve a set of algebraic equations at each step in time. In the case of linear problems this is refl ected by a need to invert a matrix at each step in time. In the case of linear and time and spatially invariant systems, this solution method requires the inversion of a matrix only once at the beginning of the problem. The difficulties cited above are no longer an impediment to problems of moderate size. In the past, explicit methods were attractive because they did not require solution of a system of algebraic equations. The disadvantage of explicit methods is that they tend to be numerically unstable, particularly when the system being investigated is lightly damped. Although MATLAB has become the computational tool of choice for systems and signal processing, the predecessors (LINPACK and EISPACK) were developed with the solution of systems of partial differential equations associated with nuclear engineering in mind. It is also interesting to note that The Math Works has

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Hamann, J., & Jacquot, R. (1996, June), Visualization Of Pde Solutions Using Implicit Methods & Matlab Paper presented at 1996 Annual Conference, Washington, District of Columbia. 10.18260/1-2--6396