Advanced Differential Equations

₹ 3,495.00

SKU: SP029 Categories: , Product ID: 1523
B.J. Gireesha, Rama S.R. Gorla, B.C. Prasannakumara
2017, Hardboundpp
xvi+596, 9789385883088
Rs. 3495.00

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