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More About This Textbook
Overview
Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.
This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing the mathematical model) and how to solve the equation (along with initial and boundary conditions). Written for advanced undergraduate and graduate students, as well as professionals working in the applied sciences, this clearly written book offers realistic, practical coverage of diffusiontype problems, hyperbolictype problems, elliptictype problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.
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Meet 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 semiindependent lessonsrather than the more usual chapterbychapter 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
Table of Contents
1. Introduction
Lesson 1. Introduction to Partial Differential Equations
2. DiffusionType Problems
Lesson 2. DiffusionType Problems (Parabolic Equations)
Lesson 3. Boundary Conditions for DiffusionType 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. HyperbolicType 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 (FourthOrder 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. FirstOrder Equations (Method of Characteristics)
Lesson 28. Nonlinear FirstOrder Equations (Conservation Equations)
Lesson 29. Systems of PDEs
Lesson 30. The Vibrating Drumhead (Wave Equation in Polar Coordinates)
4. EllipticType Problems
Lesson 31. The Laplacian (an intuitive description)
Lesson 32. General Nature of BoundaryValue 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 FiniteDifference Method
Lesson 39. An Implicit FiniteDifference Method (CrankNicolson 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 (EulerLagrange Equations)
Lesson 45. Variational Methods for Solving PDEs (Method of Ritz)
Lesson 46. Perturbation method for Solving PDEs
Lesson 47. ConformalMapping 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