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This paper addresses the critical need for the development of
intelligent or assisting software tools for the scientist who is
working in the initial problem formulation and mathematical model
representation stage of research. In particular, examples of that
representation in fluid dynamics and instability theory are
discussed. The creation of a mathematical model that is ready for
application of certain solution strategies requires extensive
symbolic manipulation of the original mathematical model. These
manipulations can be as simple as term reordering or as complicated
as discovery of various symmetry groups embodied in the equations,
whereby Backlund-type transformations create new determining
equations and integrability conditions or create differential
Grobner bases that are then solved in place of the original
nonlinear PDEs. Several examples are presented of the kinds of
problem formulations and transforms that can be frequently
encountered in model representation for fluids problems. The
capability of intelligently automating these types of transforms,
available prior to actual mathematical solution, is advocated.
Physical meaning and assumption-understanding can then be
propagated through the mathematical transformations, allowing for
explicit strategy development.
An integral part of engineering design is the development of models that describe physical behavior or phenomena in mathematical terms. As engineering systems have become more complex, classic linear methods of modeling and analysis have proved inadequate, creating a need for nonlinear models to solve design problems. This text provides an introduction to mathematical modeling of linear and nonlinear systems, with an emphasis on the solution of nonlinear design problems. While encouraging the use of the computer as a tool for modeling and analysis, the aim is to discuss the basic concepts underlying computer techniques and to seek analytical solutions. Among topics covered are exact solution, numerical solution, graphical solution, and approximate solution methods; and the stability of nonlinear systems. Numerous examples show how to apply modeling methods to real engineering systems. The book also includes end-of-chapter problems and case studies of challenging design problems. Intended for senior or beginning graduate students, this text will also serve as a helpful reference for practicing engineers.
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