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
• Initial Value Problems for Order Equations
• Nonhomogeneous equations of order
• 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
• LaguerreDifferential 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 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