A Prolog-based expert system for modeling with partial differential equations

Initial results are presented from a project to develop intelligent expert system tools for the construction and execution of com plex mathematical models which employ partial differential equations. This paper reports on that portion of the system, the expert system front end, which is concerned with the represen tation and execution of meta-level knowledge about the model building process itself. The current problem domain is restricted to models involving linear as well as a limited class of nonlinear partial differential equations of up to second order. The system is capable of examining the user's input model for inconsisten cies as well as making judgments about suitable algorithms for stable numerical solutions. The symbolic equations, along with boundary and initial conditions, are represented in the form of Prolog clauses. The input model is analyzed by the expert system for equation type as well as appropriate boundary and initial conditions. Discrete approximation of equation terms is perform ed symbolically and the results are then expanded and simplified into an efficiently computable form. Prolog clauses from the discrete approximation phase are then automatically combined with an appropriate solution stepping scheme to arrive at a fully executable algorithm for numerical solution. The algorithm may then be executed serially for testing or further processed into a form suitable for passing to a separate high-speed processor.