# A Very Applied First Course in Partial Differential Equations

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## Description

For one semester junior/senior level courses in partial differential equations in departments of Engineering, Physics, Applied Mathematics, Social Science, Biology, and other sciences.

Extending beyond the standard heat and wave equation...nwhere most other partial differential equations texts stop...nthis text spans into the spectrum of physics, engineering, mathematical physiology, environmental studies, and mathematical applications. Its "student friendly" approach will appeal to the non-math major who may be intimidated by the theoretical approach of most mathematics texts. In addition, a wide variety of real-world applications illustrate how mathematics applies directly to the non-math majors' fields of study, while giving mathematics majors the opportunity to see how mathematics relates to other courses of study.

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Extending beyond the standard heat and wave equation...nwhere most other partial differential equations texts stop...nthis text spans into the spectrum of physics, engineering, mathematical physiology, environmental studies, and mathematical applications. Its "student friendly" approach will appeal to the non-math major who may be intimidated by the theoretical approach of most mathematics texts. In addition, a wide variety of real-world applications illustrate how mathematics applies directly to the non-math majors' fields of study, while giving mathematics majors the opportunity to see how mathematics relates to other courses of study.

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## Product details

- Hardback | 507 pages
- 175.3 x 238.8 x 27.9mm | 929.88g
- 23 Jul 2001
- Pearson Education (US)
- Pearson
- United States
- English
- 0130304174
- 9780130304179

## Table of contents

Preface

1. Introduction.

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-Coefficient ODE.

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.

Bibliography.

Index.

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1. Introduction.

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-Coefficient ODE.

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.

Bibliography.

Index.

show more