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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.
As the title suggests, this book focuses on the practical: finding solutions to partial differential equations, together with some discussion on how the PDEs arise in various modeling problems. There is very little in the way of theory — few theorems regarding existence or uniqueness, no maximum principle. The book does provide a nice introduction to PDEs for junior or senior undergraduates who have had a semester of ordinary differential equations.
The book is divided into four main sections, covering parabolic, hyperbolic and elliptic equations, followed by a section on numerical methods. The topics are presented as “lessons” (47 in total), each suitable for one or possibly two class meetings. The author suggests a syllabus of 30 lessons for a one semester course, which provided a nice tour of the canonical examples and techniques. Each lesson begins with short “purpose of lesson” providing an overview of the topic at hand, and concludes with a short selection of exercises. Solutions to some of the exercises are in the appendix. Using this as a text with an audience of both mathematics and engineering majors, I found the need to supplement the exercises with additional problems.
The book covers separation of variables and a variety of integral transformation techniques. The presentation is made largely through examples. Students found the examples easy to follow and the text generally approachable. However, the lack of a theoretical overview of the techniques lead some students to have difficulty generalizing the examples. Most of lessons focus on problems with one spatial dimension; there are very few sections devoted to a higher dimensional setting. While this approach may oversimplify the material, it did allow the class to graphically explore the solution as a surface evolving through time.
This book is suitable as a text for an undergraduate introductory course in PDEs, especially one with a mixed audience of math and engineering students. A background in ordinary differential equations is a must, as the text contains little in the way of review. Most instructors will want to supplement the material in terms of both content, to add more theory, and more exercises. To this end, each lesson has a brief, albeit dated, annotated list of suggested readings.
Chris Tweddle (ct55@evansville.edu) is an Assistant Professor of Mathematics at the University of Evansville. His interests include partial differential equations (especially as tools for image processing), undergraduate research, and training pre-service teachers of mathematics.
Introduction | ||||||||
Lesson 1. | Introduction to Partial Differential Equations | |||||||
2. |
Diffusion-Type Problems |
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Lesson 2. | Diffusion-Type Problems (Parabolic Equations) | |||||||
Lesson 3. | Boundary Conditions for Diffusion-Type Problems | |||||||
Lesson 4. | Derivation of the Heat Equation | |||||||
Lesson 5. | Separation of Variables | |||||||
Lesson 6. | Transforming Nonhomogeneous BCs into Homogeneous Ones | |||||||
Lesson 7. | Solving More Complicated Problems by Separation of Variables | |||||||
Lesson 8. | Transforming Hard Equations into Easier Ones | |||||||
Lesson 9. | Solving Nonhomogeneous PDEs (Eigenfunction Expansions) | |||||||
Lesson 10. | Integral Transforms (Sine and Cosine Transforms) | |||||||
Lesson 11. | The Fourier Series and Transform | |||||||
Lesson 12. | The Fourier Transform and its Application to PDEs | |||||||
Lesson 13. | The Laplace Transform | |||||||
Lesson 14. | Duhamel's Principle | |||||||
Lesson 15. | The Convection Term u subscript x in Diffusion Problems | |||||||
3. |
Hyperbolic-Type Problems |
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Lesson 16. | The One Dimensional Wave Equation (Hyperbolic Equations) | |||||||
Lesson 17. | The D'Alembert Solution of the Wave Equation | |||||||
Lesson 18. | More on the D'Alembert Solution | |||||||
Lesson 19. | Boundary Conditions Associated with the Wave Equation | |||||||
Lesson 20. | The Finite Vibrating String (Standing Waves) | |||||||
Lesson 21. | The Vibrating Beam (Fourth-Order PDE) | |||||||
Lesson 22. | Dimensionless Problems | |||||||
Lesson 23. | Classification of PDEs (Canonical Form of the Hyperbolic Equation) | |||||||
Lesson 24. | The Wave Equation in Two and Three Dimensions (Free Space) | |||||||
Lesson 25. | The Finite Fourier Transforms (Sine and Cosine Transforms) | |||||||
Lesson 26. | Superposition (The Backbone of Linear Systems) | |||||||
Lesson 27. | First-Order Equations (Method of Characteristics) | |||||||
Lesson 28. | Nonlinear First-Order Equations (Conservation Equations) | |||||||
Lesson 29. | Systems of PDEs | |||||||
Lesson 30. | The Vibrating Drumhead (Wave Equation in Polar Coordinates) | |||||||
4. |
Elliptic-Type Problems |
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Lesson 31. | The Laplacian (an intuitive description) | |||||||
Lesson 32. | General Nature of Boundary-Value Problems | |||||||
Lesson 33. | Interior Dirichlet Problem for a Circle | |||||||
Lesson 34. | The Dirichlet Problem in an Annulus | |||||||
Lesson 35. | Laplace's Equation in Spherical Coordinates (Spherical Harmonics) | |||||||
Lesson 36. | A Nonhomogeneous Dirichlet Problem (Green's Functions) | |||||||
5. |
Numerical and Approximate Methods |
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Lesson 37. | Numerical Solutions (Elliptic Problems) | |||||||
Lesson 38. | An Explicit Finite-Difference Method | |||||||
Lesson 39. | An Implicit Finite-Difference Method (Crank-Nicolson Method) | |||||||
Lesson 40. | Analytic versus Numerical Solutions | |||||||
Lesson 41. | Classification of PDEs (Parabolic and Elliptic Equations) | |||||||
Lesson 42. | Monte Carlo Methods (An Introduction) | |||||||
Lesson 43. | Monte Carlo Solutions of Partial Differential Equations) | |||||||
Lesson 44. | Calculus of Variations (Euler-Lagrange Equations) | |||||||
Lesson 45. | Variational Methods for Solving PDEs (Method of Ritz) | |||||||
Lesson 46. | Perturbation method for Solving PDEs | |||||||
Lesson 47. | Conformal-Mapping Solution of PDEs | |||||||
Answers to Selected Problems |
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Appendix 1. | Integral Transform Tables | |||||||
Appendix 2. | PDE Crossword Puzzle | |||||||
Appendix 3. | Laplacian in Different Coordinate Systems | |||||||
Appendix 4. | Types of Partial Differential Equations | |||||||
Index |