Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems

In this undergraduate/graduate textbook, the authors introduce ODEs and PDEs through 50 class-tested lectures. Mathematical concepts are explained with clarity and rigor, using fully worked-out examples and helpful illustrations. Exercises are provided at the end of each chapter for practice. The treatment of ODEs is developed in conjunction with PDEs and is aimed mainly towards applications. The book covers important applications-oriented topics such as solutions of ODEs in form of power series, special functions, Bessel functions, hypergeometric functions, orthogonal functions and polynomials, Legendre, Chebyshev, Hermite, and Laguerre polynomials, theory of Fourier series.

Undergraduate and graduate students in mathematics, physics and engineering will benefit from this book. The book assumes familiarity with calculus.

Solvable Differential Equations.- Second-Order Differential Equations.- Preliminaries to Series Solutions.- Solution at an Ordinary Point.- Solution at a Singular Point.- Solution at a Singular Point (Cont’d.).- Legendre Polynomials and Functions.- Chebyshev, Hermite and Laguerre Polynomials.- Bessel Functions.- Hypergeometric Functions.- Piecewise Continuous and Periodic Functions.- Orthogonal Functions and Polynomials.- Orthogonal Functions and Polynomials (Cont’d.).- Boundary Value Problems.- Boundary Value Problems (Cont’d.).- Green’s Functions.- Regular Perturbations.- Singular Perturbations.- Sturm–Liouville Problems.- Eigenfunction Expansions.- Eigenfunction Expansions (Cont’d.).- Convergence of the Fourier Series.- Convergence of the Fourier Series (Cont’d.).- Fourier Series Solutions of Ordinary Differential Equations.- Partial Differential Equations.- First-Order Partial Differential Equations.- Solvable Partial Differential Equations.- The Canonical Forms.- The Method of Separation of Variables.- The One-Dimensional Heat Equation.- The One-Dimensional Heat Equation (Cont’d.).- The One-Dimensional Wave Equation.- The One-Dimensional Wave Equation (Cont’d.).- Laplace Equation in Two Dimensions.- Laplace Equation in Polar Coordinates.- Two-Dimensional Heat Equation.- Two-Dimensional Wave Equation.- Laplace Equation in Three Dimensions.- Laplace Equation in Three Dimensions (Cont’d.).- Nonhomogeneous Equations.- Fourier Integral and Transforms.- Fourier Integral and Transforms (Cont’d.).- Fourier Transform Method for Partial DEs.- Fourier Transform Method for Partial DEs (Cont’d.).- Laplace Transforms.- Laplace Transforms (Cont’d.).- Laplace Transform Method for Ordinary DEs.- Laplace Transform Method for Partial DEs.- Well-PosedProblems.- Verification of Solutions.