INTRODUCTION TO MATHEMATICA Introduction Conventions Getting Started File Manipulation Differential Equations To the Instructor To the Student MathSource INTRODUCTION Notation and Definitions Initial and Boundary Conditions Classification of Second Order Equations Some Known Equations Superposition Principle METHOD OF CHARACTERISTICS First Order Equations Linear Equations with Constant Coefficients Linear Equations with Variable Coefficients First Order Quasi-Linear Equations First Order Nonlinear Equations Geometrical Considerations Some Theorems on Characteristics Second Order Equations Linear and Quasi-linear Equations LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS Inverse Operators Homogeneous Equations Nonhomogeneous Equations ORTHOGONAL EXPANSIONS Orthogonality Orthogonal Polynomials Series of Orthogonal Functions Trigonometric Fourier Series Eigenfunction Expansions Bessel Functions SEPARATION OF VARIABLES Introduction Hyperbolic Equations Parabolic Equations Elliptic Equations Cylindrical Coordinates Spherical Coordinates Nonhomogeneous Problems INTEGRAL TRANSFORMS Laplace Transforms Notation Basic Laplace Transforms Inversion Theorem Fourier Transforms Fourier Integral Theorems Properties of Fourier Transforms Fourier Sine and Cosine Transforms Finite Fourier Transforms GREEN''S FUNCTIONS Generalized Functions Green''s Functions Elliptic Equations Parabolic Equations Hyperbolic Equations Applications of Green''s Functions Computation of Green''s Functions BOUNDARY VALUE PROBLEMS Initial and Boundary Conditions Implicit Conditions Periodic Conditions Wave Propagation and Dispersion Boundary Layer Flows Miscellaneous Problems WEIGHTED RESIDUAL METHODS Line Integrals Variational Notation Multiple Integrals Weak Variational Formulation Gauss-Jacobi Quadrature Rayleigh-Ritz Method Choice of Test Functions Transient Problems Other Methods PERTURBATION METHODS Perturbation Problem Taylor Series Expansions Successive Approximations Boundary Perturbations Fluctuating Flows FINITE DIFFERENCE METHODS Finite Difference Schemes First Order Equations Second Order Equations Appendix A: Green''s Identities Appendix B: Orthogonal Polynomials Appendix C: Tables of Transform Pairs Appendix D: Glossary of Mathematica Functions Appendix E: Mathematica Packages and Notebooks Bibliography Index Each chapter also contains sections of Mathematica Projects and Exercisest;BR>ORTHOGONAL EXPANSIONS Orthogonality Orthogonal Polynomials Series of Orthogonal Functions Trigonometric Fourier Series Eigenfunction Expansions Bessel Functions SEPARATION OF VARIABLES Introduction Hyperbolic Equations Parabolic Equations Elliptic Equations Cylindrical Coordinates Spherical Coordinates Nonhomogeneous Problems INTEGRAL TRANSFORMS Laplace Transforms Notation Basic Laplace Transforms Inversion Theorem Fourier Transforms Fourier Integral Theorems Properties of Fourier Transforms Fourier Sine and Cosine Transforms Finite Fourier Transforms GREEN''S FUNCTIONS Generalized Functions Green''s Functions Elliptic Equations Parabolic Equations Hyperbolic Equations Applications of Green''s Functions Computation of Green''s Functions BOUNDARY VALUE PROBLEMS Initial and Boundary Conditions Implicit Conditions Periodic Conditions Wave Propagation and Dispersion Boundary Layer Flows Miscellaneous Problems WEIGHTED RESIDUAL METHODS Line Integrals Variational Notation Multiple Integrals Weak Variational Formulation Gauss-Jacobi Quadrature Rayleigh-Ritz Method Choice of Test Functions Transient Problems Other Methods PERTURBATION METHODS Perturbation Problem Taylor Series Expansions Successive Approximations Boundary Perturbations Fluctuating Flows FINITE DIFFERENCE METHODS Finite Difference Schemes First Order Equations Second Order Equations Appendix A: Green''s Identities Appendix B: Orthogonal Polynomials Appendix C: Tables of Transform Pairs Appendix D: Glossary of Mathematica Functions Appendix E: Mathematica Packages and Notebooks Bibliography Index Each chapter also contains sections of Mathematica Projects and Exercisess Computation of Green''s Functions BOUNDARY VALUE PROBLEMS Initial and Boundary Conditions Implicit Conditions Periodic Conditions Wave Propagation and Dispersion Boundary Layer Flows Miscellaneous Problems WEIGHTED RESIDUAL METHODS Line Integrals Variational Notation Multiple Integrals Weak Variational Formulation Gauss-Jacobi Quadrature Rayleigh-Ritz Method Choice of Test Functions Transient Problems Other Methods PERTURBATION METHODS Perturbation Problem Taylor Series Expansions Successive Approximations Boundary Perturbations Fluctuating Flows FINITE DIFFERENCE METHODS Finite Difference Schemes First Order Equations Second Order Equations Appendix A: Green''s Identities Appendix B: Orthogonal Polynomials Appendix C: Tables of Transform Pairs Appendix D: Glossary of Mathematica Functions Appendix E: Mathematica Packages and Notebooks Bibliography Index Each chapter also contains sections of Mathematica Projects and Exercisesynomials Appendix C: Tables of Transform Pairs Appendix D: Glossary of Mathematica Functions Appendix E: Mathematica Packages and Notebooks Bibliography Index Each chapter also contains sections of Mathematica Projects and Exercises.