A Very Applied First Course in Partial Differential Equations / Edition 1by Michael K. Keane
Pub. Date: 07/28/2001
Publisher: Prentice Hall
This extremely readable book illustrates how mathematics applies directly to different fields of study. Focuses on problems that require physical to mathematical translations, by showing readers how equations have actual meaning in the real world. Covers fourier integrals, and transform methods, classical PDE problems, the Sturm-Liouville Eigenvalue problem, and much more. For readers interested in partial differential equations.
- Prentice Hall
- Publication date:
- Product dimensions:
- 7.33(w) x 9.23(h) x 1.00(d)
Table of ContentsPreface
2. The One-Dimensional Heat Equation.
Introduction. Derivation of Heat Conduction in a One-Dimensional Rod. Boundary Conditions for a One-Dimensional Rod. The Maximum Principle and Uniqueness. Steady-State Temperature Distribution.
3. The One-Dimensional Wave Equation.
Introduction. Derivation of the One-Dimensional Wave Equation. Boundary Conditions. Conservation of Energy For a Vibrating String. Method of Characteristics. D'Alembert's Solution to the One-Dimensional Wave Equation.
4. The Essentials of Fourier Series.
Introduction. Elements of Linear Algebra. A New Space: The Function Space of Piecewise Smooth Functions. Even and Odd Functions and Fourier Series.
5. Separation of Variables: The Homogeneous Problem.
Introduction. Operators: Linear and Homogeneous Equations. Separation of Variables: Heat Equation. Separation of Variables: Wave Equation. The Multidimensional Spatial Problem. Laplace's Equation.
6. The Calculus of Fourier Series.
Introduction. Fourier Series Representation of a Function: Fourier Series As a Function. Differentiation of Fourier Series. Integration of Fourier Series. Fourier Series and the Gibbs Phenomenon.
7. Separation of Variables: The Nonhomogeneous Problem.
Introduction. Nonhomogeneous PDEs With Homogeneous BCS. Homogeneous PDE With Nonhomogeneous BCS. Nonhomogeneous PDE and BCS. Summary.
8. The Sturm-Liouville Eigenvalue Problem.
Introduction. Definition of the Sturm-Liouville Eigenvalue Problem. Rayleigh Quotient. The General PDE Example. Problems Involving Homogeneous BCS of the Third Kind.
9. Solution of Linear Homogeneous Variable-CoefficientODE.
Introduction. Some Facts About the General Second-Order Ode. Euler's Equation. Brief Review of Power Series. The Power Series Solution Method. Legender's Equation and Legendre Polynomials. Method of Frobenius and Bessel's Equation.
10. Classical PDE Problems.
Introduction. Laplace's Equation. Transverse Vibrations of a Thin Beam. Heat Conduction in a Circular Plate. Schrodinger's Equation. The Telegrapher's Equation. Interesting Problems in Diffusion.
11. Fourier Integrals and Transform Methods.
Introduction. The Fourier Integral. The Laplace Transform. The Fourier Transform. Fourier Transform Solution Method.
Appendix A: Summary of the Spatial Problem.
Appendix B: Proofs of Related Theorems.
Theorems from Chapter 2. Theorems from Chapter 4. Theorem from Chapter 5.
Appendix C: Basics From Ordinary Differential Equations.
Some Solution Methods for First-Order Odes. Some Solution Methods of Second-Order Odes.
Appendix D: Mathematical Notation.
Appendix E: Summary of Thermal Diffusivity of Common Materials.
Appendix F: Tables of Fourier and Laplace Transforms.
Tables of Fourier, Fourier Cosine, and Fourier Sine Transforms. Table of Laplace Transforms.
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