Systematic Imposition of Partial Differential Equations in Boundary Value Problems

We present a systematic approach to impose the partial differential equations of second-order boundary value problems. This is motivated by the needs of scientific computing and software development. The contemporary need is for software which does not restrict end users to some pre-given equations. For this we first generalize the idea of a field as a formal sum of differential forms on a Minkowski manifold. Thereafter for these formal sums we write in space-time first-order partial differential equations that stem with the action principle. Since boundary value problems are solved with respect to some spatial and temporal chart, i.e., coordinate system, the corresponding differential operator is written explicitly in space and time. By construction, particular fields, such as those of electromagnetism or elasticity become instances of the formal sums, and the differential equations that constitute the underlying field theories become instances of the corresponding differential operator. Consequently, the approach covers a wide class of field models and yields a systematic approach to impose specific partial differential equations. As it is straightforward to convert the underlying operators to pieces of software, the approach yields solid foundations to develop software systems that are not restricted to pre-given equations.