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# Boundary Value Problems

## 2nd Edition

Author:
eBook ISBN: 9781483269788
Published Date: 1st January 1979
Page Count: 366
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## Description

Boundary Value Problems is a text material on partial differential equations that teaches solutions of boundary value problems. The book also aims to build up intuition about how the solution of a problem should behave.

The text consists of seven chapters. Chapter 1 covers the important topics of Fourier Series and Integrals. The second chapter deals with the heat equation, introducing separation of variables. Material on boundary conditions and Sturm-Liouville systems is included here. Chapter 3 presents the wave equation; estimation of eigenvalues by the Rayleigh quotient is mentioned briefly. The potential equation is the topic of Chapter 4, which closes with a section on classification of partial differential equations. Chapter 5 briefly covers multidimensional problems and special functions. The last two chapters, Laplace Transforms and Numerical Methods, are discussed in detail.

The book is intended for third and fourth year physics and engineering students.

1 Fourier Series and Integrals

1 Periodic Functions and Fourier Series

2 Arbitrary Period. Periodic Extensions

3 Even and Odd Functions. Half-Range Expansions

4 Convergence of Fourier Series

5 Uniform Convergence

6 Summability of Fourier Series

7 Mean Error and Convergence in Mean

8 Numerical Determination of Fourier Coefficients

9 Complex Methods

10 Fourier Integral

11 Applications of Fourier Series and Integrals

2 The Heat Equation

1 Derivation and Boundary Conditions

2 Example: Fixed End Temperatures

3 Example: Insulated Bar

4 Example: Convection

5 Sturm-Liouville Problems

6 Expansion in Series of Eigenfunctions

7 Generalities on the Heat Conduction Problem

8 Semi-Infinite Rod

9 Infinite Rod

3 The Wave Equation

1 The Vibrating String

2 Solution of the Vibrating String Problem

3 D'Alembert's Solution

4 Generalities on the One-Dimensional Wave Equation

5 Estimation of Eigenvalues

6 Wave Equation in Unbounded Regions

7 References

4 The Potential Equation

1 Potential Equation

2 Potential in a Rectangle

3 Potential in a Slot

4 Potential in a Disk

5 Classification of Partial Differential Equations and Limitations of the Product Method

5 Problems in Several Dimensions

1 Derivation of the Two-Dimensional Wave Equation

2 Derivation of the Two-Dimensional Heat Equation

3 Solution of the Two-Dimensional Heat Equation

4 Problems in Polar Coordinates

5 Bessel's Equation

6 Vibrations of a Circular Membrane

7 Some Applications of Bessel Functions

8 Spherical Coordinates. Legendre Polynomials

9 References

6 Laplace Transform

1 Definition and Elementary Properties

2 Elementary Applications. Partial Fractions and Convolutions

3 Applications to Partial Differential Equations

4 More Difficult Examples

7 Numerical Methods

1 Boundary Value Problems

2 The Heat Equation

3 Two-Dimensional Heat Equation

4 Potential Equation

5 References

Bibliography

Index

No. of pages:
366
Language:
English