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

ISBN-13: 9780387791456 / Angielski / Miękka / 2008 / 410 str.

Ravi P. Agarwal; Donal O'Regan

Opis

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.

Recenzje

From the reviews:

"This work by Agarwal (Florida Institute of Technology) and O'Regan ... gives a clear introduction to the fields of study identified in its title. ... The authors provide 50 short lectures on various topics in the theory of ordinary and partial differential equations. This format makes independent reading of the book easier since it condenses the material into sections of manageable length. ... Beginning graduate students would be the ideal audience for such self-study. Summing Up: Recommended. Academic audiences, upper-division undergraduates and above." (S. L. Sullivan, Choice, Vol. 47 (1), September, 2009)

"The book comprises 50 class-tested lectures which both the authors have given to engineering and mathematics major students under the titles Boundary Value Problems and Methods of Mathematical Physics at various institutions all over the globe ... . The prerequisite for this book is calculus, so it can be used for a senior undergraduate course. It should also be suitable for a beginning graduate course ... . The answers and hints to almost all the exercises are provided for the convenience of the reader." (Juri M. Rappoport, Zentralblatt MATH, Vol. 1172, 2009)

Spis treści

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-Posed Problems.- Verification of Solutions.

Blurb

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 80^th 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.