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 diffusion-type problems, hyperbolic-type problems, elliptic-type problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.
Stanley J. Farlow
Country of Publication:
Series: Dover Books on Mathematics
01 September 1993
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