Introduction |
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Lesson 1. |
Introduction to Partial Differential Equations |
2.
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Diffusion-Type Problems
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Lesson 2. |
Diffusion-Type Problems (Parabolic Equations) |
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Lesson 3. |
Boundary Conditions for Diffusion-Type Problems |
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Lesson 4. |
Derivation of the Heat Equation |
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Lesson 5. |
Separation of Variables |
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Lesson 6. |
Transforming Nonhomogeneous BCs into Homogeneous Ones |
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Lesson 7. |
Solving More Complicated Problems by Separation of Variables |
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Lesson 8. |
Transforming Hard Equations into Easier Ones |
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Lesson 9. |
Solving Nonhomogeneous PDEs (Eigenfunction Expansions) |
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Lesson 10. |
Integral Transforms (Sine and Cosine Transforms) |
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Lesson 11. |
The Fourier Series and Transform |
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Lesson 12. |
The Fourier Transform and its Application to PDEs |
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Lesson 13. |
The Laplace Transform |
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Lesson 14. |
Duhamel's Principle |
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Lesson 15. |
The Convection Term u subscript x in Diffusion Problems |
3.
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Hyperbolic-Type Problems
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Lesson 16. |
The One Dimensional Wave Equation (Hyperbolic Equations) |
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Lesson 17. |
The D'Alembert Solution of the Wave Equation |
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Lesson 18. |
More on the D'Alembert Solution |
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Lesson 19. |
Boundary Conditions Associated with the Wave Equation |
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Lesson 20. |
The Finite Vibrating String (Standing Waves) |
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Lesson 21. |
The Vibrating Beam (Fourth-Order PDE) |
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Lesson 22. |
Dimensionless Problems |
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Lesson 23. |
Classification of PDEs (Canonical Form of the Hyperbolic Equation) |
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Lesson 24. |
The Wave Equation in Two and Three Dimensions (Free Space) |
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Lesson 25. |
The Finite Fourier Transforms (Sine and Cosine Transforms) |
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Lesson 26. |
Superposition (The Backbone of Linear Systems) |
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Lesson 27. |
First-Order Equations (Method of Characteristics) |
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Lesson 28. |
Nonlinear First-Order Equations (Conservation Equations) |
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Lesson 29. |
Systems of PDEs |
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Lesson 30. |
The Vibrating Drumhead (Wave Equation in Polar Coordinates) |
4.
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Elliptic-Type Problems
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Lesson 31. |
The Laplacian (an intuitive description) |
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Lesson 32. |
General Nature of Boundary-Value Problems |
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Lesson 33. |
Interior Dirichlet Problem for a Circle |
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Lesson 34. |
The Dirichlet Problem in an Annulus |
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Lesson 35. |
Laplace's Equation in Spherical Coordinates (Spherical Harmonics) |
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Lesson 36. |
A Nonhomogeneous Dirichlet Problem (Green's Functions) |
5.
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Numerical and Approximate Methods
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Lesson 37. |
Numerical Solutions (Elliptic Problems) |
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Lesson 38. |
An Explicit Finite-Difference Method |
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Lesson 39. |
An Implicit Finite-Difference Method (Crank-Nicolson Method) |
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Lesson 40. |
Analytic versus Numerical Solutions |
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Lesson 41. |
Classification of PDEs (Parabolic and Elliptic Equations) |
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Lesson 42. |
Monte Carlo Methods (An Introduction) |
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Lesson 43. |
Monte Carlo Solutions of Partial Differential Equations) |
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Lesson 44. |
Calculus of Variations (Euler-Lagrange Equations) |
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Lesson 45. |
Variational Methods for Solving PDEs (Method of Ritz) |
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Lesson 46. |
Perturbation method for Solving PDEs |
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Lesson 47. |
Conformal-Mapping Solution of PDEs |
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Answers to Selected Problems
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Appendix 1. |
Integral Transform Tables |
Appendix 2. |
PDE Crossword Puzzle |
Appendix 3. |
Laplacian in Different Coordinate Systems |
Appendix 4. |
Types of Partial Differential Equations |
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Index |