Search

Browse Subject Areas

For Authors

Submit a Proposal

Ordinary Differential Equations and Special Functions

By Dipankar De
Copyright: 2025   |   Expected Pub Date:2025//
ISBN: 9781394385034  |  Hardcover  |  
639 pages
Price: $225 USD
Add To Cart

One Line Description
This book is an essential guide for anyone in engineering or mathematical physics looking to master the fundamental concepts of differential equations and special functions, which are crucial for solving real-world problems.

Audience
Academics, researchers, physicists, mathematicians, and engineers studying the applications of advanced equations

Description
In today’s evolving mathematics landscape, differential equations and special functions have shown great promise for applications in engineering. Problems in mathematical physics help determine solutions for differential equations under certain parameters, which can be turned into new special functions, such as Bessel’s functions, to measure electricity, hydrodynamics, and vibration. Ordinary Differential Equations and Special Functions serves as a fundamental guide to these concepts, covering everything from elementary-level special functions to differential equations with a series of solutions.

Back to Top
Author / Editor Details
Dipankar De, PhD is a former contractual professor and guest lecturer in an Indian University and now a retired Associate Professor with more than 40 years of experience. He has published several research papers in various reputed journals in the fields of fuzzy mathematics and differential geometry.

Back to Top

Table of Contents
Preface
About the Book
Introduction
Part I: Ordinary Differential Equations
1. Preliminaries I
1.1 Introduction
1.2 Formation of a Differential Equation
1.3 Family of Curves Represented by Ordinary Differential Equations
1.4 Equation of the First Order and First Degree
1.4.1 Variable Seperable
1.4.2 Homogeneous Equations
1.4.3 Linear Equation
1.4.4 Bernoulli’s Equation
1.4.5 Exact Differential Equation
1.5 Equations of the First Order and Higher Degree
1.5.1 Clairaut’s Equation
1.6 Linear Differential Equation
1.6.1 The General Form
1.6.2 Rules for Finding the Complementary Function
1.6.3 Symbolic Operator 1 f(D)
1.6.4 Rules for Finding the Particular Integral
1.7 Other Methods of Finding P.I.
1.7.1 Method of Variation of Parameters
1.7.2 Cauchy’s Homogeneous Linear Equation
1.8 Differential Equation of Other Types
1.8.1 Equations Whose One Solution is Known
1.8.2 Reduction to Normal Form by Removing the First Derivative
1.8.3 Change of Independent Variable
1.8.4 Total Differential Equations
1.9 Orthogonal Trajectories
1.10 Examples
1.11 Exercise
2. Existence Theorems
2.1 Introduction
2.2 Initial Value Problems and Boundary Value Problems
2.3 Picard’s Method of Successive Approximation
2.3.1 Picard’s Method for Solving Simultaneous Differential Equations with Initial Conditions
2.4 Lipschitz Condition
2.5 Picard’s Theorem: Existence and Uniqueness Theorem
2.5.1 Picard’s Theorem
2.6 Singular Solutions
2.6.1 Theory of Singular Solutions
2.7 Clairaut Equation
2.8 Examples
2.9 Exercise
3. Systems of Linear Differential Equations-I
3.1 Introduction
3.2 Matrix Form of a Linear System
3.3 Reduction of an nth-Order Equation
3.4 Matrix Preliminaries
3.5 Fundamental Set of Solutions
3.5.1 Existence and Uniqueness
3.5.2 Solution Set
3.5.3 Fundamental Matrix
3.6 Solution of Non-Homogeneous Linear Systems
3.7 Linear System with Constant Coefficients
3.8 Exercise
4. Systems of Linear Differential Equations-II
4.1 Introduction
4.2 Linearly Dependent and Independent Functions
4.3 The Second-Order Homogeneous Equation
4.3.1 Initial Value Problems for Second-Order Equations
4.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters
4.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients
4.5.1 Initial Value Problems for nth-Order Equations
4.5.2 The n-th Order Non-Homogeneous Equation
4.5.3 A Special Method for Solving the Non-Homogeneous Equation
4.6 Examples
4.7 Exercise
5. Adjoint Equation
5.1 Introduction
5.2 Adjoint Equation
5.2.1 Lagrange Identity
5.2.2 Self-Adjoint Equation
5.2.3 Adjoint Equations for Linear Equations of Order One
5.2.4 Adjoint Equations for Linear Equations of Order Two
5.3 Green’s Formula
5.4 Examples
5.5 Exercise
6. Boundary Value Problem
6.1 Introduction
6.2 Green’s Function
6.3 Examples
6.4 Exercise
7. Strum Liouville Problem
7.1 Introduction
7.2 Strum–Liouville Equation
7.3 Orthogonality of Eigen Functions
7.4 Orthonormal Set of Functions
7.5 Gram–Schmidt Process of Orthonormalization
7.6 Reality of Eigenvalues
7.7 Examples
7.8 Exercise
Part II: Special Functions
8. Preliminaries II
8.1 Introduction
8.2 Infinite Series
8.2.1 Convergent and Divergent Series
8.2.2 Alternating Series
8.2.3 Uniform Convergence
8.2.4 Convergent Test
8.2.5 Properties of Uniformly Convergent Series
8.3 Infinite Integrals
8.4 Infinite Products
8.5 Some Theorems on Functions of Complex Variables
8.6 Exercise
9. Series Solution of Differential Equations
9.1 Introduction
9.2 Power Series
9.3 Power Series Solution Near the Ordinary Point x = x0
9.4 Series Solution About Regular Singular Point x = 0: Frobenius Method
9.4.1 Method of Differentiation
9.5 Examples
9.6 Exercise
10. Hypergeometric Functions
10.1 Introduction
10.1.1 Factorial Function
10.1.2 Hypergeometric Series
10.1.3 Symmetric Property of Hypergeometric Functions
10.2 Differentiation of Hypergeometric Functions
10.3 An Integral Formula for a Hypergeometric Function
10.4 Transformation of F (α,β ,γ ;x)
10.4.1 Simple Transformations
10.4.2 Quadratic Transformations
10.5 Hypergeometric Equation
10.6 Confluent Hypergeometric Series
10.6.1 Solution of the Confluent Hypergeometric Differential Equation when x = 0 and γ is not an integer
10.6.2 Differentiation of Confluent Hypergeometric Functions
10.6.3 Integral Representation for Confluent Hypergeometric Function
10.6.4 Theorem (Kummer’s)
10.7 Contiguous Hypergeometric Functions
10.7.1 Contiguity Relationship
10.8 Generalized Hypergeometric Series
10.8.1 Convergence of a Generalized Hypergeometric Function
10.8.2 Differential Equations Satisfied by pFq
10.8.3 Properties of pFq
10.9 Integrals Involving Generalized Hypergeometric Functions
10.10 Some Special Generalized Hypergeometric Functions
10.11 Barnes Type Contour Integrals
10.12 Example
10.13 Exercise
11. Bessel Functions
11.1 Introduction
11.2 Bessel’s Equation
11.3 Recurrence Formulae for Jn(x)
11.4 Expansion of J0 , J1 ,and J1 2
11.5 Generating Function for Jn(x)
11.6 Modified Bessel Functions
11.6.1 Recurrence Relations for Modified Bessel Functions
11.7 Equations Reducible to Bessel Equation
11.8 Orthogonality of Bessel Functions
11.8.1 Fourier–Bessel Expansion for f(x)
11.9 Zeros of Bessel Functions
11.10 Ber and Bei Functions
11.11 Exercise
12. Legendre Polynomials
12.1 Introduction
12.2 Legendre’s Equation
12.3 Another Form of Legendre’s Polynomial Pn(x)
12.3.1 Legendre’s Function of the First Kind or Legendre’s Polynomial of Degree n
12.3.2 Legendre’s Function of the Second Kind
12.3.3 Determination of the First Few Legender’s Polynomials Pn(x)
12.4 Generating Function for Legendre’s Polynomials
12.5 Various Forms of Pn(x)
12.5.1 Rodrigues’ Formula
12.5.2 Murphy’s Formula
12.5.3 Bateman’s Generating Relations
12.6 Recurrence Formulae for Pn(x)
12.7 Christoffel’s Summation Formula
12.7.1 Christoffel’s Expansion
12.8 Orthogonality of Legendre Polynomials
12.9 Fourier–Legendre’s Expansion of f (x)
12.9.1 Expansion of xn in Legendre Polynomials
12.10 Associated Legendre’s Functions
12.10.1 Properties of the Associated Legendre’s Functions
12.10.2 Orthogonality Relation for Pn x m( )
12.10.3 Recurrence Relations for Pn x P m r ( ) m
12.11 Legendre’s Functions of the Second Kind—Qn(x)
12.11.1 Some Useful Results
12.11.2 Recurrence Relations for Qn(x)
12.11.3 Relation of Pn(x) and Qn(x)
12.12 Examples
12.13 Exercise
13. Hermite Polynomials
13.1 Introduction
13.2 Hermite Equation and Its Solution
13.3 Generating Function for Hermite Polynomials
13.4 Recurrence Relations
13.4.1 Some Results
13.5 Orthogonal Property
13.6 Expansion of Polynomials
13.7 More Generating Functions
13.8 Examples
13.9 Exercise
14. Laguerre Polynomials
14.1 Introduction
14.2 Laguerre’s Equation and Its Solution
14.3 Generating Function of Laguerre Polynomials
14.3.1 Alternative Expression of Laguerre Polynomials
14.3.2 First Few Laguerre Polynomials
14.4 Orthogonality Properties of Laguerre Polynomials
14.5 Recurrence Relations
14.6 Expansion of Laguerre Polynomials
14.7 Properties of Laguerre Polynomials
14.7.1 Differential Equation of Ln(x)
14.7.2 Integral Properties
14.8 Generalized Laguerre Polynomial
14.8.1 Introduction
14.8.2 Recurrence Relations
14.8.3 Rodrigues Formula
14.8.4 Expansions
14.8.5 Some Special Results
14.9 Examples
14.10 Exercise
15. Jacobi Polynomials
15.1 Introduction
15.2 Jacobi Polynomial
15.3 Generating Functions
15.4 Rodrigues’ Formula
15.5 Orthogonality of Jacobi Polynomial
15.5.1 Properties of Orthogonality
15.6 Recurrence Relations
15.6.1 (a) Relations in which shift n is involved
15.6.2 (b) Relations in which shifts in α ,β ,and n are involved
15.7 Expansions
15.8 Examples
15.9 Exercise
16. Chebyshev Polynomials
16.1 Introduction
16.2 Chebyshev Polynomials
16.3 Orthogonality Property
16.4 Recurrence Relations
16.5 Identities of Chebyshev Polynomials
16.6 Expansions
16.7 Generating Function
16.8 Rodrigues Formula of Chebyshev Polynomials
16.8.1 Special Values of Tn(x)
16.9 Exercise
Appendix A: Answer to Even-Numbered Exercises
References
Index

Back to Top



Description
Author/Editor Details
Table of Contents
Bookmark this page