Some Basic Results Euclidean Space Classes of Continuous Functions Convergence Functionals Linear Transformations Cramer''s Rule Green''s Identities Differentiation and Integration Inequalities The Concept of Green''s Functions Generalized Functions Singular Distributions The Concept of Green''s Functions Linear Operators and Inverse Operators Fundamental Solutions Sturm-Liouville Systems Ordinary Differential Equations Initial Value Problems Boundary Value Problems Eigenvalue Problem for Sturm-Liouville Systems Periodic Sturm-Liouville Systems Singular Sturm-Liouville Systems Bernoulli''s Separation Method Coordinate Systems Partial Differential Equations Bernoulli''s Separation Method Examples Integral Transforms Integral Transform Pairs Laplace Transform Fourier Integral Theorems Fourier Sine and Cosine Transforms Finite Fourier Transforms Multiple Transforms Hankel Transforms Summary: Variables of Transforms Parabolic Equations 1-D Diffusion Equation 2-D Diffusion Equation 3-D Diffusion Equation Schrödinger Diffusion Operator Min-Max Principle Diffusion Equation in a Finite Medium Axisymmetric Diffusion Equation 1-D Heat Conduction Problem Stefan Problem 1-D Fractional Diffusion Equation 1-D Fractional Schrödinger Diffusion Equation Eigenpairs and Dirac Delta Function Hyperbolic Equations 1-D Wave Equation 2-D Wave Equation 3-D Wave Equation 2-D Axisymmetric Wave Equation Vibrations of a Circular Membrane 3-D Wave Equation in a Cube Schrödinger Wave Equation Hydrogen Atom 1-D Fractional Nonhomogeneous Wave Equation Applications of the Wave Operator Laplace Transform Method Quasioptics and Diffraction Elliptic Equations Green''s Function for 2-D Laplace''s Equation 2-D Laplace''s Equation in a Rectangle Green''s Function for 3-D Laplace''s Equation Harmonic Functions 2-D Helmholtz''s Equation Green''s Function for 3-D Helmholtz''s Equation 2-D Poisson''s Equation in a Circle Method for Green''s Function in a Rectangle Poisson''s Equation in a Cube Laplace''s Equation in a Sphere Poisson''s Equation and Green''s Function in a Sphere Applications of Elliptic Equations Spherical Harmonics Historical Sketch Laplace''s Solid Spherical Harmonics Surface Spherical Harmonics Conformal Mapping Method Definitions and Theorems Dirichlet Problem Neumann Problem Green''s and Neumann''s Functions Computation of Green''s Functions Appendix A: Adjoint Operators Appendix B: List of Fundamental Solutions Appendix C: List of Spherical Harmonics Appendix D: Tables of Integral Transforms Appendix E: Fractional Derivatives Appendix F: Systems of Ordinary Differential Equations Bibliography Index Exercises appear at the end of each chapter, with hints, answers, and, sometimes, complete solutions. STRONG>Bernoulli''s Separation Method Coordinate Systems Partial Differential Equations Bernoulli''s Separation Method Examples Integral Transforms Integral Transform Pairs Laplace Transform Fourier Integral Theorems Fourier Sine and Cosine Transforms Finite Fourier Transforms Multiple Transforms Hankel Transforms Summary: Variables of Transforms Parabolic Equations 1-D Diffusion Equation 2-D Diffusion Equation 3-D Diffusion Equation Schrödinger Diffusion Operator Min-Max Principle Diffusion Equation in a Finite Medium Axisymmetric Diffusion Equation 1-D Heat Conduction Problem Stefan Problem 1-D Fractional Diffusion Equation 1-D Fractional Schrödinger Diffusion Equation Eigenpairs and Dirac Delta Function Hyperbolic Equations 1-D Wave Equation 2-D Wave Equation 3-D Wave Equation 2-D Axisymmetric Wave Equation Vibrations of a Circular Membrane 3-D Wave Equation in a Cube Schrödinger Wave Equation Hydrogen Atom 1-D Fractional Nonhomogeneous Wave Equation Applications of the Wave Operator Laplace Transform Method Quasioptics and Diffraction Elliptic Equations Green''s Function for 2-D Laplace''s Equation 2-D Laplace''s Equation in a Rectangle Green''s Function for 3-D Laplace''s Equation Harmonic Functions 2-D Helmholtz''s Equation Green''s Function for 3-D Helmholtz''s Equation 2-D Poisson''s Equation in a Circle Method for Green''s Function in a Rectangle Poisson''s Equation in a Cube Laplace''s Equation in a Sphere Poisson''s Equation and Green''s Function in a Sphere Applications of Elliptic Equations Spherical Harmonics Historical Sketch Laplace''s Solid Spherical Harmonics Surface Spherical Harmonics Conformal Mapping Method Definitions and Theorems Dirichlet Problem Neumann Problem Green''s and Neumann''s Functions Computation of Green''s Functions Appendix A: Adjoint Operators Appendix B: List of Fundamental Solutions Appendix C: List of Spherical Harmonics Appendix D: Tables of Integral Transforms Appendix E: Fractional Derivatives Appendix F: Systems of Ordinary Differential Equations Bibliography Index Exercises appear at the end of each chapter, with hints, answers, and, sometimes, complete solutions. ion 1-D Fractional Schrödinger Diffusion Equation Eigenpairs and Dirac Delta Function Hyperbolic Equations 1-D Wave Equation 2-D Wave Equation 3-D Wave Equation 2-D Axisymmetric Wave Equation Vibrations of a Circular Membrane 3-D Wave Equation in a Cube Schrödinger Wave Equation Hydrogen Atom 1-D Fractional Nonhomogeneous Wave Equation Applications of the Wave Operator Laplace Transform Method Quasioptics and Diffraction Elliptic Equations Green''s Function for 2-D Laplace''s Equation 2-D Laplace''s Equation in a Rectangle Green''s Function for 3-D Laplace''s Equation Harmonic Functions 2-D Helmholtz''s Equation Green''s Function for 3-D Helmholtz''s Equation 2-D Poisson''s Equation in a Circle Method for Green''s Function in a Rectangle Poisson''s Equation in a Cube Laplace''s Equation in a Sphere Poisson''s Equation and Green''s Function in a Sphere Applications of Elliptic Equations Spherical Harmonics Historical Sketch Laplace''s Solid Spherical Harmonics Surface Spherical Harmonics Conformal Mapping Method Definitions and Theorems Dirichlet Problem Neumann Problem Green''s and Neumann''s Functions Computation of Green''s Functions Appendix A: Adjoint Operators Appendix B: List of Fundamental Solutions Appendix C: List of Spherical Harmonics Appendix D: Tables of Integral Transforms Appendix E: Fractional Derivatives Appendix F: Systems of Ordinary Differential Equations Bibliography Index Exercises appear at the end of each chapter, with hints, answers, and, sometimes, complete solutions. t;BR>2-D Poisson''s Equation in a Circle Method for Green''s Function in a Rectangle Poisson''s Equation in a Cube Laplace''s Equation in a Sphere Poisson''s Equation and Green''s Function in a Sphere Applications of Elliptic Equations Spherical Harmonics Historical Sketch Laplace''s Solid Spherical Harmonics Surface Spherical Harmonics Conformal Mapping Method Definitions and Theorems Dirichlet Problem Neumann Problem Green''s and Neumann''s Functions Computation of Green''s Functions Appendix A: Adjoint Operators Appendix B: List of Fundamental Solutions Appendix C: List of Spherical Harmonics Appendix D: Tables of Integral Transforms Appendix E: Fractional Derivatives Appendix F: Systems of Ordinary Differential Equations Bibliography Index Exercises appear at the end of each chapter, with hints, answers, and, sometimes, complete solutions. ppendix D: Tables of Integral Transforms Appendix E: Fractional Derivatives Appendix F: Systems of Ordinary Differential Equations Bibliography Index Exercises appear at the end of each chapter, with hints, answers, and, sometimes, complete solutions.