Mathematics in Physics and Engineering

Contents Preface Chapter I. Introduction To Partial Differential Equations 1. Introduction 2. The One-Dimensional Wave Equation 3. Method Of Separation Of Variables 4. The Two-Dimensional Wave Equation 5. Three-Dimensional Wave Equation 6. The Wave Equation In Plane And Cylindrical Polar Coordinates A. Plane Polars B. Cylindrical Polars 7. The Wave Equation In Spherical Polar Coordinates 8. Laplace's Equation In Two Dimensions A. Cartesian Coordinates B. Polar Coordinates 9. Laplace's Equation In Three Dimensions 10. The Diffusion Or Heat Flow Equation 10.1. Neutron Diffusion 11. A Fourth Order Partial Differential Equation 12. The Bending Of An Elastic Plate - The Biharmonic Equation 13. Characteristics 13.1. Cauchy's Problem 13.2. Reduction Of (13.1.1) To The Standard Form 13.3. Riemann's Method Of Solution Of (13.1.1) 13.4. Numerical Integration Of Hyperbolic Differential Equations Problems General References Chapter II. Ordinary Differential Equations: Frobenius' And Other Methods Of Solution 1. Introduction 2. Solution In Series By The Method Of Frobenius 3. Bessel's Equation 4. Legendre's Equation 5. Hyper Geometric Equation 6. Series Solution About A Point Other Than The Origin 6.1. The Transformation X = (1 - ¿)/2 7. Series Solution In Descending Powers Of X 8. Confluent Hypergeometric Equation 8.1. Laguerre Polynomials 8.2. Hermite Polynomials 9. Asymptotic Or Semi-Convergent Series 10. Change Of Dependent Variable 11. Change Of The Independent Variable 12. Exact Equations 13. The Inhomogeneous Linear Equation 14. Perturbation Theory For Non-Linear Differential Equations 14.1. The Perturbation Method 14.2. Periodic Solutions Problems General References Chapter III. Bessel And Legendre Functions 1. Definition Of Special Functions 127 2. Jn(X), The Bessel Function Of The First Kind Of Order N 2.1. Recurrence Relations: Jn(X) 3. Bessel Function Of The Second Kind Of Order N, Yn(X) 4. Equations Reducible To Bessel's Equation 5. Applications 6. Modified Bessel Functions: In(X), Kn(X) 6.1. Recurrence Relations For In(X) And Kn(X) 6.2. Equations Reducible To Bessel's Modified Equation 6.3. Bessel Functions Of The Third Kind (Hankel Functions) 7. Illustrations Involving Modified Bessel Functions 8. Orthogonal Properties 8.1. Expansion Of F(X) In Terms Of Jn(¿ix) 8.2. Jn(X) As An Integral (Where N Is Zero Or An Integer) 8.3. Other Important Integrals 9. Integrals Involving The Modified Bessel Functions 10. Zeros Of The Bessel Functions 11. A Generating Function For The Legendre Polynomials 11.1. Recurrence Relations 11.2. Orthogonality Relations For The Legendre Polynomials 11.3. Associated Legendre Functions 12. Applications From Electromagnetism 13. Spherical Harmonics 14. The Addition Theorem For Spherical Harmonics Problems General References Chapter IV. The Laplace And Other Transforms 1. Introduction 2. Laplace Transforms And Some General Properties 3. Solution Of Linear Differential Equations With Constant Coefficients 4. Further Theorems And Their Application 5. Solution Of The Equation F(D)x(t) = F(t) By Means Of The Convolution Theorem 6. Application To Partial Differential Equations 7. The Finite Sine Transform 8. The Simply Supported Rectangular Plate 9. Free Oscillations Of A Rectangular Plate 10. Plate Subject To Combined Lateral Load And A Uniform Compression 11.