Partial Differential Equations for Scientists and Engineers (Dover Books on Advanced Mathematics)

  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