Computer facilitated generalized coordinate transformations of partial differential equations with engineering applications

Abstract

Partial differential equations (PDEs) play an important role in describing many physical, industrial, and biological processes. Their solutions could be considerably facilitated by using appropriate coordinate transformations. There are many coordinate systems besides the well-known Cartesian, polar, and spherical coordinates. In this article, we illustrate how to make such transformations using Maple. Such a use has the advantage of easing the manipulation and derivation of analytical expressions. We illustrate this by considering a number of engineering problems governed by PDEs in different coordinate systems such as the bipolar, elliptic cylindrical, and prolate spheroidal. In our opinion, the use of Maple or similar computer algebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommon transformations more frequently at the very least for situations where the transformations provide smarter and easier solutions.

Using a free open source software to teach mathematics

“…In this article, we show how one can use an open source CAS to create teaching scenarios similar to those developed by other authors using alternative systems (e.g., Refs. ).…”