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An Expert System for Partial Differential Equations

Abstract

The execution of the solution, by the separation of variables process, of the Poisson, diffusion, and wave equations (homogeneous or nonhomogeneous) in rectangular, cylindrical, or spherical coordinate systems, with Dirichlet, Neumann, Robin, singular, periodic, or Sommerfeld boundary conditions, can be carried out in the time, Laplace, or frequency domains by a decision-tree process, using a library of eigenfunctions. We describe an expert system, USFKAD, that has been constructed for this purpose.

Partial Differential Equations in the Engineering Curriculum

Every practicing engineer whose specialty involves modeling of physical phenomena, such as electromagnetic fields, temperature, sound, stress and strain, fluid flow, diffusion, etc., has to deal with the mathematical syntax of the discipline - the partial differential equation (PDE). For example, the electrical engineering undergraduate lecturer in electromagnetics, semiconductor processing, thermal issues in electronic packaging, etc. should be able to call on this mathematical concept, at least peripherally, to provide the students some familiarity with the technical issues involved in the quantitative models. However, this subject (PDEs) is vast and complicated, and compromises have to be made in incorporating it into the undergraduate's curriculum. A 2-semester course that deals honestly and rigorously with the subject is out of the question.

The compromises presently employed in engineering programs at undergraduate institutions are:

(1) A short treatment of PDEs that relies completely on numerical solvers; or

(2) A brief tutorial that covers the basics of the separation of variables technique.

Each of these is unsatisfactory. (1) is inferior to (2) because, even with the graphic capabilities of today's hardware and software, it is extremely difficult for an inexperienced undergraduate user to tell, from a vast assemblage of tabulations and graphs, how the solutions will respond to changes in the boundary conditions or the physical dimensions - issues of prime importance to engineering. For example consider the frequency of the resonant mode of a rectangular cavity with sides X, Y and Z given by ω = cπ 1/ X 2 + 1/ Y 2 + 1/ Z 2 . This is not a terribly complicated formula, but contemplate trying to deduce it from graphs!

The eigenfunction expansions yielded by (2) do reveal these dependencies (and are exact). The drawback of this solution procedure is the lack of time to impart expertise in its implementation except for a few elementary cases - rectangular geometries and ideal conductors, for instance.

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Snider, A., & Kadamani, S. (2006, June), An Expert System For Teaching Partial Differential Equations Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--51