Applied Mathematics for Science and Engineering

1Problem Formulation and Model Development

Introduction

Our purpose in this course is to review some mathematical techniques that can be used to solve important problems in engineering and the applied sciences. We will focus on problem types that are crucial to the analysis and simulation of real, physical phenomena. Sometimes, our objective will be to predict the future behavior of a system and sometimes it will be to interpret behavior that has already occurred. We want to stress that the author and the readers are collaborators in this effort, and whether this text is being used in a formal setting or for self-study, the ultimate goal is the same: We want to be able deal with problems that arise in the applied sciences and do so efficiently. And—this is important—we do not want to rely on calculation software unless we know something about the method(s) being employed. Too often, real problems can have multiple solutions, so it is essential that the analyst be able to exercise some judgment based on understanding of the problem and of the algorithm that has been selected.

Many of the problems we will be solving will come from both transient and equilibrium balances, and they will involve forces, fluxes, and the couplings between driving force–flux pairs. Examples of the latter are Newton's, Fourier's, and Fick's laws:

(1.1)

where τ_yx is the shear stress (acting on a y-plane due ...