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 