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Partial Differential Equations for Scientists and Engineers

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Table of Contents

1. Introduction Lesson 1. Introduction to Partial Differential Equations 2. Diffusion-Type Problems 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 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 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 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 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

About the Author

Partial Differential Equations & Beyond Stanley J. Farlow's Partial Differential Equations for Scientists and Engineers is one of the most widely used textbooks that Dover has ever published. Readers of the many Amazon reviews will easily find out why. Jerry, as Professor Farlow is known to the mathematical community, has written many other fine texts - on calculus, finite mathematics, modeling, and other topics. We followed up the 1993 Dover edition of the partial differential equations title in 2006 with a new edition of his An Introduction to Differential Equations and Their Applications. Readers who wonder if mathematicians have a sense of humor might search the internet for a copy of Jerry's The Girl Who Ate Equations for Breakfast (Aardvark Press, 1998). Critical Acclaim for Partial Differential Equations for Scientists and Engineers: "This book is primarily intended for students in areas other than mathematics who are studying partial differential equations at the undergraduate level. The book is unusual in that the material is organized into 47 semi-independent lessonsrather than the more usual chapter-by-chapter approach. "An appealing feature of the book is the way in which the purpose of each lesson is clearly stated at the outset while the student will find the problems placed at the end of each lesson particularly helpful. The first appendix consists of integral transform tables whereas the second is in the form of a crossword puzzle which the diligent student should be able to complete after a thorough reading of the text. "Students (and teachers) in this area will find the book useful as the subject matter is clearly explained. The author and publishers are to be complimented for the quality of presentation of the material." - K. Morgan, University College, Swansea

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