An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation

Book Preface

This book is suitable for courses in applied mathematics with numerics, basic fluid mechanics, basic mathematics of electromagnetism, or mathematical modeling. The prerequisites for students are vector calculus, basic differential equations, the rudiments of matrix algebra, knowledge of some programming language, and of course some mathematical maturity. No knowledge of partial differential equations or numerical analysis is assumed.

The author has used parts of this book while teaching courses in mathematical modeling at the University of Missouri where students (undergraduate and graduate) of engineering, the sciences, and mathematics enrolled. This heterogeneous mix of students should be expected in a course at the advanced undergraduate beginning graduate level with a title such as Mathematical Modeling I. Thus, the instructor must assess the abilities and background knowledge of the students who show up on the first day of class. Professors should be prepared and willing to modify their syllabus after a week or two of instruction to accommodate their students. In fact, the most likely modification is to cover less material at a slower pace. Perhaps learning a few concepts and techniques well is always more valuable than exposure to a survey of new ideas.

A typical 15-week semester course might consist of one lecture on Chapter 1, two weeks on Chapter 2 (mostly ODE), two weeks on Chapter 5 (fundamental physical modeling, reaction-diffusion systems, and basic numerics for simple parabolic PDE), one week on Chapter 6 (electrical signals on neurons and traveling wave solutions), and one week on Chapter 8 (basic PID control) to complete approximately half of the semester. Of course only parts of the material in these chapters (in particular Chapter 5) can be covered in detail in class. By this time in the semester at least three substantial homework assignments should be completed using exercises, problems, and projects suggested in the text. Of course, there is good reason to also include exercises designed by the instructor. At least, students should have written, tested, and reported applications to applied problems of a few basic codes for approximating solutions of ODEs and PDEs. Their work should be presented in (carefully) written reports (in English prose [or some other language]) where analysis and discussion of results are supplemented

with references to output from numerical experiments in tabular or graphical formats. In-class exams are possible but perhaps not as appropriate to the material as homework assignments. The book does not contain many routine problems; in fact, many problems and all of the projects are open ended. How else will students experience challenges that anticipate realistic applied problems? Some of the projects introduce new concepts and are fleshed out accordingly. A list of suggested projects is given in the index (see the entry Projects). The second half of the semester might be devoted to continuum mechanics or electromagnetism. But, the usual choice is fluid mechanics. There will be sufficient time to derive the conservation of momentum equation and discuss the Euler and Navier–Stokes stress tensors as in Chapter 11. Standard applications include flow in a pipe (Chapter 12) followed by a discussion of potential flow with applications to circulation, lift, and drag in Chapter 13. Perhaps the end of the semester is reached with a discussion of the Coriolis effect on drains and hurricanes. The final exam can be replaced by a set of problems and projects taken from Chapters 10 and 19, with respect given to sufficient background material discussed in class. In addition, each student might be required to present a project—in the spirit of the course—taken directly from this book, related to their work in some other class, or related to their research.

A more advanced course might be devoted entirely to continuum mechanics with the intention of covering more sophisticated mathematics and numerics. In particular, basic water wave phenomena and free-surface flow can be addressed along with appropriate numerical methods. In Chapter 16, a complete treatment of Chorin’s projection method is given in sufficient detail for students (and perhaps their professor) to write a basic CFD code that can be applied to a diverse set of applied problems. This is followed by the most mathematically sophisticated part of the book on the boundary element method, where classical potential theory is covered and all the ingredients of this numerical method are discussed in detail. This is followed by a treatment of smoothed particle hydrodynamics, again with sufficient detail to write a viable code. Channel flow provides a modeling experience along with a discussion and application of Prandtl’s boundary layer theory, and a solid treatment of the theory and numerics of hyperbolic conservation laws. All of this material is written in context with applied problems. The chapter ends with a basic discussion of elastic solids, continuum mechanics, the weak formulation of PDEs, and sufficient detail to write a basic finitexvi
element code that can be used to approximate the solutions of problems that arise in modeling elastic solids.

Likewise, an advanced course might be devoted to applied problems in electromagnetism. The material in Chapter 20 provides a basic (mathematically oriented) introduction to Maxwell’s equations and the electromagnetic boundary value problem. An enlightening application of the theory is made to transverse electromagnetic waves and waveguides. This is specialized to the theory of transmission lines where the Riemann problem for hyperbolic conservation laws arises in context and its solution is used to construct a viable numerical method to approximate the electromagnetic waves. This theory is applied to the practical problem of time-domain reflectometry, which serves as an introduction to a basic inverse problem of wide interest: shine radiation on some object with the intent of identifying the object by analyzing the reflected electromagnetic waves.

The material in the book can be used to design undergraduate research projects and master’s projects. Of course, it can also be used to help PhD students gain valuable experience before approaching an applied research problem.