Systematic derivation of partial differential equations for second order boundary value problems

Abstract

Software systems designed to solve second order boundary value problems are typically restricted to hardwired lists of partial differential equations. In order to come up with more flexible systems, we introduce a systematic approach to find partial differential equations that result in eligible boundary value problems. This enables one to construct and combine one's own partial differential equations instead of choosing those from a pre-given list. This expands significantly end users possibilities to employ boundary value problems in modeling. To introduce the main ideas we employ differential geometry to examine the mathematical structure involved in second order boundary value problems and exploit electromagnetism as a working example. This provides us with an organized view on the key building blocks behind boundary value problems. Thereafter the approach is naturally generalized to a class of second order boundary value problems that covers field theories from statics to wave problems. As a result, we obtain a systematic framework to construct partial differential equations and to test whether they form eligible boundary value problems.