## About the Book

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.

**B.J. Gireesha** is an Assistant Professor, Department of Mathematics, Kuvempu University, Shimoga, Karnataka, India.

**Rama S.R. Gorla** is Professor Emeritus, Department of Mechanical Engineering, Cleveland State University, Cleveland, Ohio 44115 USA, and currently, Department of Mechanical Engineering, University of Akron, Akron, Ohio 44325 USA.

**B.C. Prasannakumara** is Faculty of Mathematics, GFGC, Koppa, Karnataka, India.

**1. First Order Linear Differential Equations **

Introduction

First Order Linear Differential Equations

Separable Differential Equations

Exact Differential Equations

Bernoulli Differential Equation

Method of Substitutions

Applications

Exercises

**2. Higher Order Differential Equations**

Introduction

The Second Order Homogeneous Differential Equations

Initial Value Problem

Linear Dependence and Independence

Second Order Nonhomogeneous Differential Equations

Linear homogeneous equations of order n

Initial Value Problems for n^th Order Equations

Nonhomogeneous equations of order n

Linear Equations with Variable Coefficients

Exercises

**3. Oscillations of Second Order Differential Equations**

Introduction

Fundamental Results

Boundary Value Problem

Green’s Function

Applications

Exercises

**4. Solutions in Terms of Power Series**

Introduction

Legendre Differential Equation

Hermite Differential Equation

Regular Singular Points

Laguerre Differential Equation

Chebyshev Differential Equation

Hypergeometric Equation

Bessel’s Equation

Applications

Exercises

**5. Successive Approximation Theory**

Introduction

Solution by Successive Approximations

Lipschitz Condition

Convergence of the Successive Approximations

Exercises

**6. System of Ordinary Differential Equations**

Introduction

Linear System of Ordinary Differential Equations

Homogeneous Linear System of Differential Equations

Nonhomogeneous Linear Systems

System of Nonlinear Differential Equations

Exercises

**7. First Order Partial Differential Equations**

Introduction

Construction of First-order Partial Differential Equations

Solutions of First Order Partial Differential Equations

Solutions Using Charpit’s Method

Method of Cauchy Characteristics

Method of Separation of Variables

Applications

Exercises

**8. Second Order Partial Differential Equations**

Introduction

Origin of Second Order Equations

Linear Partial Differential Equations with Constant Coefficients

Equations with Variable Coefficients

Canonical Forms

Classification of Second-Order Equations in n Variables

Modeling with Second Order Equations

Exercises

**9. Parabolic Equations**

Introduction

Solutions by Separation of Variables

Solutions by Eigenfunction Expansion Method

Solutions by Laplace Transform Method

Solutions by Fourier Transforms Method

Duhamel’s Principle

Similarity Transformation Method

Solutions to Higher Dimensional Equations

Solutions to Miscellaneous Problems

Exercises

**10. Hyperbolic Equations**

Introduction

Method of Characteristics (D’Alembert Solution)

Solutions by Separation of Variables

Solutions by Eigenfunctions Expansion Method

Solutions by Laplace Transform Method

Solutions by Fourier Transform Method

Duhamel’s Principle

Similarity Transformation Method

Solutions to Higher Dimensional Equations

Solutions to Miscellaneous Problems

Exercises

**11. Elliptic Equations**

Introduction

Solutions by Separation of Variables

Solutions by Eigenfunctions Expansion Method

Solutions by Fourier Transform Method

Similarity Transformation Method

Solutions to Higher Dimensional Equations

Solutions to Miscellaneous Problems

Exercises

**12. Perturbation Theory**

Introduction

Perturbation Theory

Algebraic Equations

Boundary Layer Problems

Boundary Layer Problems for Linear Ordinary Differential Equations

Method of Multiple Scales

Eigenvalues of Perturbed Problems

Solutions of Differential Equations by Regular Perturbation

Solutions of Partial Differential Equations

Inner and Outer Expansions in Partial Differential Equations (Matched Asymptotic Expansions)

Analysis and Improvement of Perturbation Series

Improvement of the Series

Exercises

**Answers to the Selected Problems**

Bibliography

Index