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

Ravi P. Agarwal, Donal O’Regan (auth.)0387791450, 9780387791456

This textbook provides a genuine treatment of ordinary and partial differential equations (ODEs and PDEs) through 50 class tested lectures.

Key Features:
Explains mathematical concepts with clarity and rigor, using fully worked-out examples and helpful illustrations.Develops ODEs in conjuction with PDEs and is aimed mainly toward applications.Covers importat applications-oriented topics such as solutions of ODEs in the form of power series, special functions, Bessel functions, hypergeometric functions, orthogonal functions and polynomicals, Legendre, Chebyshev, Hermite, and Laguerre polynomials, and the theory of Fourier series.Provides exercises at the end of each chapter for practice.
This book is ideal for an undergratuate or first year graduate-level course, depending on the university. Prerequisites include a course in calculus.

About the Authors:

Ravi P. Agarwal received his Ph.D. in mathematics from the Indian Institute of Technology, Madras, India. He is a professor of mathematics at the Florida Institute of Technology. His research interests include numerical analysis, inequalities, fixed point theorems, and differential and difference equations. He is the author/co-author of over 800 journal articles and more than 20 books, and actively contributes to over 40 journals and book series in various capacities.

Donal O’Regan received his Ph.D. in mathematics from Oregon State University, Oregon, U.S.A. He is a professor of mathematics at the National University of Ireland, Galway. He is the author/co-author of 15 books and has published over 650 papers on fixed point theory, operator, integral, differential and difference equations. He serves on the editorial board of many mathematical journals.

Previously, the authors have co-authored/co-edited the following books with Springer: Infinite Interval Problems for Differential, Difference and Integral Equations; Singular Differential and Integral Equations with Applications; Nonlinear Analysis and Applications: To V. Lakshmikanthan on his 80th Birthday; An Introduction to Ordinary Differential Equations.

In addition, they have collaborated with others on the following titles: Positive Solutions of Differential, Difference and Integral Equations; Oscillation Theory for Difference and Functional Differential Equations; Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations.

Table of contents :
Front Matter….Pages I-XIV
Solvable Differential Equations….Pages 1-7
Second-Order Differential Equations….Pages 8-14
Preliminaries to Series Solutions….Pages 15-22
Solution at an Ordinary Point….Pages 23-30
Solution at a Singular Point….Pages 31-36
Solution at a Singular Point (Cont’d.)….Pages 37-46
Legendre Polynomials and Functions….Pages 47-56
Chebyshev, Hermite and Laguerre Polynomials….Pages 57-63
Bessel Functions….Pages 64-74
Hypergeometric Functions….Pages 75-82
Piecewise Continuous and Periodic Functions….Pages 83-89
Orthogonal Functions and Polynomials….Pages 90-94
Orthogonal Functions and Polynomials (Cont’d.)….Pages 95-103
Boundary Value Problems….Pages 104-108
Boundary Value Problems (Cont’d.)….Pages 109-118
Green’s Functions….Pages 119-128
Regular Perturbations….Pages 129-137
Singular Perturbations….Pages 138-144
Sturm–Liouville Problems….Pages 145-156
Eigenfunction Expansions….Pages 157-162
Eigenfunction Expansions (Cont’d.)….Pages 163-170
Convergence of the Fourier Series….Pages 171-175
Convergence of the Fourier Series (Cont’d.)….Pages 176-186
Fourier Series Solutions of Ordinary Differential Equations….Pages 187-193
Partial Differential Equations….Pages 194-201
First-Order Partial Differential Equations….Pages 202-209
Solvable Partial Differential Equations….Pages 210-218
The Canonical Forms….Pages 219-226
The Method of Separation of Variables….Pages 227-233
The One-Dimensional Heat Equation….Pages 234-240
The One-Dimensional Heat Equation (Cont’d.)….Pages 241-248
The One-Dimensional Wave Equation….Pages 249-255
The One-Dimensional Wave Equation (Cont’d.)….Pages 256-265
Laplace Equation in Two Dimensions….Pages 266-274
Laplace Equation in Polar Coordinates….Pages 275-283
Two-Dimensional Heat Equation….Pages 284-291
Two-Dimensional Wave Equation….Pages 292-299
Laplace Equation in Three Dimensions….Pages 300-305
Laplace Equation in Three Dimensions (Cont’d.)….Pages 306-315
Nonhomogeneous Equations….Pages 316-322
Fourier Integral and Transforms….Pages 323-329
Fourier Integral and Transforms (Cont’d.)….Pages 330-337
Fourier Transform Method for Partial DEs….Pages 338-343
Fourier Transform Method for Partial DEs (Cont’d.)….Pages 344-353
Laplace Transforms….Pages 354-360
Laplace Transforms (Cont’d.)….Pages 361-373
Laplace Transform Method for Ordinary DEs….Pages 374-383
Laplace Transform Method for Partial DEs….Pages 384-393
Well-Posed Problems….Pages 394-398
Verification of Solutions….Pages 399-404
Back Matter….Pages 405-410