FlexPDE is a software tool for finding numerical solutions to systems of linear or non-linear partial differential equations using the methods of finite element analysis. The systems may represent static boundary value, time dependent initial/boundary value, or eigenvalue problems. Rather than addressing the solution of specific equations related to a given area of application, FlexPDE provides a framework for treating partial differential equation systems in general. It gives users a straightforward method of defining the equations, domains and boundary conditions appropriate to their application. From this description it creates a finite element solution process tailored to the problem. Within quite broad limits, then, FlexPDE is able to construct a numerical solution to a wide range of applications, without itself having any built-in knowledge of any of them.
The goal of this technical note is not to provide a discussion of the specific grammatical rules of writing scripts for FlexPDE, nor to describe the operation of the graphical user interface. Those topics are covered in other volumes of the FlexPDE documentation, the Getting Started guide, the User Guide tutorial, and the Problem Descriptor Reference.
We will address several fields of physics in which FlexPDE finds fruitful application, describing the various problems, the mathematical statement of the partial differential equation system, and the ultimate posing of the problem to FlexPDE. The volume is accompanied by the text of all the examples, which the user can submit to FlexPDE to see the solution in progress or use as a foundation for problems of his own.
This manual is emphatically not a compendium of the problems FlexPDE “knows how to solve”. It is rather a group of examples showing ways in which the power of FlexPDE can be applied to partial differential equations systems in many fields. The true range of applicability of FlexPDE can be demonstrated only by the full range of ingenuity of users with insight into the mathematics of their own special fields.
Nor does this manual attempt to present textbook coverage of the theory of the topics addressed. The range of applications addressable by FlexPDE would make such an attempt impossible, even if we were capable of such an endeavor. Instead, we have presented enough of the theory of each topic to allow those practitioners who are familiar with the subject to see how the material has been analyzed and presented to FlexPDE. Users who are unfamiliar with the various fields of application should consult standard textbooks to find the full theoretical development of the subjects.