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| Preface | |
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| Basic Review | |
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| Preparation for Maple Worksheets | |
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| Preparation for Linear Algebra | |
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| Preparation for Ordinary Differential Equations | |
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| Preparation for Partial Differential Equations | |
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| Ordinary Linear Differential Equations | |
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| Introduction | |
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| First-Order Linear Differential Equations | |
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| First-Order Initial-Value Problem | |
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| Second-Order Linear Differential Equations with Constant Coefficients | |
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| Second-Order Linear Differential Equations with Variable Coefficients | |
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| Finding a Second Basis Vector by the Method of Reduction of Order | |
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| The Method of Variation of Parameters-Second-Order Green's Function | |
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| Initial-Value Problem for Second-Order Differential Equations | |
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| Frobenius Method of Series Solutions to Ordinary Differential Equations | |
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| Series Sine and Cosine Solutions to the Euler Differential Equation | |
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| Frobenius Series Solution to the Bessel Differential Equation | |
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| Chapter Summary | |
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| Exercises | |
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| Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series | |
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| Introduction | |
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| The Regular Sturm-Liouville Eigenvalue Problem | |
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| Green's Formula and the Statement of Orthonormality | |
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| The Generalized Fourier Series Expansion | |
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| Examples of Regular Sturm-Liouville Eigenvalue Problems | |
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| Nonregular or Singular Sturm-Liouville Eigenvalue Problems | |
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| Chapter Summary | |
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| Exercises | |
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| The Diffusion or Heat Partial Differential Equation | |
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| Introduction | |
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| One-Dimensional Diffusion Operator in Rectangular Coordinates | |
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| Method of Separation of Variables for the Diffusion Equation | |
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| Sturm-Liouville Problem for the Diffusion Equation | |
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| Initial Conditions for the Diffusion Equation in Rectangular Coordinates | |
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| Example Diffusion Problems in Rectangular Coordinates | |
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| Verification of Solutions-Three-Step Verification Procedure | |
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| Diffusion Equation in the Cylindrical Coordinate System | |
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| Initial Conditions for the Diffusion Equation in Cylindrical Coordinates | |
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| Example Diffusion Problems in Cylindrical Coordinates | |
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| Chapter Summary | |
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| Exercises | |
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| The Wave Partial Differential Equation | |
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| Introduction | |
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| One-Dimensional Wave Operator in Rectangular Coordinates | |
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| Method of Separation of Variables for the Wave Equation | |
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| Sturm-Liouville Problem for the Wave Equation | |
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| Initial Conditions for the Wave Equation in Rectangular Coordinates | |
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| Example Wave Equation Problems in Rectangular Coordinates | |
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| Wave Equation in the Cylindrical Coordinate System | |
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| Initial Conditions for the Wave Equation in Cylindrical Coordinates | |
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| Example Wave Equation Problems in Cylindrical Coordinates | |
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| Chapter Summary | |
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| Exercises | |
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| The Laplace Partial Differential Equation | |
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| Introduction | |
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| Laplace Equation in the Rectangular Coordinate System | |
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| Sturm-Liouville Problem for the Laplace Equation in Rectangular Coordinates | |
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| Example Laplace Problems in the Rectangular Coordinate System | |
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| Laplace Equation in Cylindrical Coordinates | |
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| Sturm-Liouville Problem for the Laplace Equation in Cylindrical Coordinates | |
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| Example Laplace Problems in the Cylindrical Coordinate System | |
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| Chapter Summary | |
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| Exercises | |
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| The Diffusion Equation in Two Spatial Dimensions | |
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| Introduction | |
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| Two-Dimensional Diffusion Operator in Rectangular Coordinates | |
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| Method of Separation of Variables for the Diffusion Equation in Two Dimensions | |
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| Sturm-Liouville Problem for the Diffusion Equation in Two Dimensions | |
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| Initial Conditions for the Diffusion Equation in Rectangular Coordinates | |
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| Example Diffusion Problems in Rectangular Coordinates | |
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| Diffusion Equation in the Cylindrical Coordinate System | |
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| Initial Conditions for the Diffusion Equation in Cylindrical Coordinates | |
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| Example Diffusion Problems in Cylindrical Coordinates | |
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| Chapter Summary | |
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| Exercises | |
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| The Wave Equation in Two Spatial Dimensions | |
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| Introduction | |
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| Two-Dimensional Wave Operator in Rectangular Coordinates | |
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| Method of Separation of Variables for the Wave Equation | |
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| Sturm-Liouville Problem for the Wave Equation in Two Dimensions | |
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| Initial Conditions for the Wave Equation in Rectangular Coordinates | |
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| Example Wave Equation Problems in Rectangular Coordinates | |
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| Wave Equation in the Cylindrical Coordinate System | |
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| Initial Conditions for the Wave Equation in Cylindrical Coordinates | |
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| Example Wave Equation Problems in Cylindrical Coordinates | |
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| Chapter Summary | |
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| Exercises | |
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| Nonhomogeneous Partial Differential Equations | |
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| Introduction | |
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| Nonhomogeneous Diffusion or Heat Equation | |
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| Initial Condition Considerations for the Nonhomogeneous Heat Equation | |
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| Example Nonhomogeneous Problems for the Diffusion Equation | |
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| Nonhomogeneous Wave Equation | |
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| Initial Condition Considerations for the Nonhomogeneous Wave Equation | |
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| Example Nonhomogeneous Problems for the Wave Equation | |
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| Chapter Summary | |
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| Exercises | |
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| Infinite and Semi-infinite Spatial Domains | |
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| Introduction | |
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| Fourier Integral | |
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| Fourier Sine and Cosine Integrals | |
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| Nonhomogeneous Diffusion Equation over Infinite Domains | |
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| Convolution Integral Solution for the Diffusion Equation | |
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| Nonhomogeneous Diffusion Equation over Semi-infinite Domains | |
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| Example Diffusion Problems over Infinite and Semi-infinite Domains | |
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| Nonhomogeneous Wave Equation over Infinite Domains | |
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| Wave Equation over Semi-infinite Domains | |
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| Example Wave Equation Problems over Infinite and Semi-infinite Domains | |
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| Laplace Equation over Infinite and Semi-infinite Domains | |
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| Example Laplace Equation over Infinite and Semi-infinite Domains | |
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| Chapter Summary | |
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| Exercises | |
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| Laplace Transform Methods for Partial Differential Equations | |
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| Introduction | |
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| Laplace Transform Operator | |
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| Inverse Transform and Convolution Integral | |
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| Laplace Transform Procedures on the Diffusion Equation | |
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| Example Laplace Transform Problems for the Diffusion Equation | |
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| Laplace Transform Procedures on the Wave Equation | |
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| Example Laplace Transform Problems for the Wave Equation | |
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| Chapter Summary | |
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| Exercises | |
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| References | |
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| Index | |