Advanced Differential Equations

Contents: 1. First order linear differential equations. 2. Higher order differential equations. 3. Oscillations of second order differential equations. 4. Solutions in terms of power series. 5. Successive approximation theory. 6. System of ordinary differential equations. 7. First order partial differential equations. 8. Second order partial differential equations. 9. Parabolic equations. 10. Hyperbolic equations. 11. Elliptic equations. 12. Perturbation theory. Answers to the selected problems. Bibliography. Index. The subject of differential equations is playing a very important role in engineering and sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern techniques of applied mathematics in modeling physical phenomena. This renewal of interest, both in research and teaching, has led to the writing of the present book. The purpose of this textbook on advanced differential equations is to meet the current and future needs of these advances in scientific developments. This text book is a detailed analysis of both ordinary and partial differential equations for students who have completed Calculus. It provides the motivation, mathematical analysis, and physical application of mathematical models in understanding practical problems in science and engineering. This book is written in such a way as to establish the mathematical ideas underlying model development leading to specific practical applications. This book consists of 12 chapters; first six chapters are devoted to study ordinary differential equations of first order linear/nonlinear, second order and higher order differential equations. Special functions, oscillations of second order differential equations, Sturm Liouville boundary value problems, Construction of Greens function are also studied. Finally the system of linear/nonlinear differential equations and autonomous systems, critical points and their stability analysis is included. The last six chapters deal with partial differential equations. Chapter 7 gives an introduction to partial differential equations. In Chapter 8, we have covered formation of second order partial differential equations and the solution of equations having constant coefficients and also the canonical forms. In the next three chapters, we have covered Fourier series, Laplace transform, Fourier transform in unbounded regions, similarity solutions, boundary value problems in rectangular, cylindrical and spherical coordinates and developed Bessel and Legendre functions. The last chapter deals with perturbation solutions of partial differential equations which have applications in science and engineering.…