Difference Equations Theory, Applications and Advanced Topics 3rd Edition by Ronald E. Mickens

Book Description

Difference Equations: Theory, Applications and Advanced Topics, Third Edition provides a broad introduction to the mathematics of difference equations and some of their applications. Many worked examples illustrate how to calculate both exact and approximate solutions to special classes of difference equations. Along with adding several advanced topics, this edition continues to cover general, linear, first-, second-, and n-th order difference equations; nonlinear equations that may be reduced to linear equations; and partial difference equations.

New to the Third Edition

• New chapter on special topics, including discrete Cauchy–Euler equations; gamma, beta, and digamma functions; Lambert W-function; Euler polynomials; functional equations; and exact discretizations of differential equations
• New chapter on the application of difference equations to complex problems arising in the mathematical modeling of phenomena in engineering and the natural and social sciences
• Additional problems in all chapters
• Expanded bibliography to include recently published texts related to the subject of difference equations

Suitable for self-study or as the main text for courses on difference equations, this book helps readers understand the fundamental concepts and procedures of difference equations. It uses an informal presentation style, avoiding the minutia of detailed proofs and formal explanations.

Table of Contents

THE DIFFERENCE CALCULUS
GENESIS OF DIFFERENCE EQUATIONS
DEFINITIONS
DERIVATION OF DIFFERENCE EQUATIONS
EXISTENCE AND UNIQUENESS THEOREM
OPERATORS ∆ AND E
ELEMENTARY DIFFERENCE OPERATORS
FACTORIAL POLYNOMIALS
OPERATOR ∆−1 AND THE SUM CALCULUS

FIRST-ORDER DIFFERENCE EQUATIONS
INTRODUCTION
GENERAL LINEAR EQUATION
CONTINUED FRACTIONS
A GENERAL FIRST-ORDER EQUATION: GEOMETRICAL METHODS
A GENERAL FIRST-ORDER EQUATION: EXPANSION TECHNIQUES

LINEAR DIFFERENCE EQUATIONS
INTRODUCTION
LINEARLY INDEPENDENT FUNCTIONS
FUNDAMENTAL THEOREMS FOR HOMOGENEOUS EQUATIONS
INHOMOGENEOUS EQUATIONS
SECOND-ORDER EQUATIONS
STURM–LIOUVILLE DIFFERENCE EQUATIONS

LINEAR DIFFERENCE EQUATIONS
INTRODUCTION
HOMOGENEOUS EQUATIONS
CONSTRUCTION OF A DIFFERENCE EQUATION HAVING SPECIFIED SOLUTIONS
RELATIONSHIP BETWEEN LINEAR DIFFERENCE AND DIFFERENTIAL EQUATIONS
INHOMOGENEOUS EQUATIONS: METHOD OF UNDETERMINED COEFFICIENTS
INHOMOGENEOUS EQUATIONS: OPERATOR METHODS
z-TRANSFORM METHOD
SYSTEMS OF DIFFERENCE EQUATIONS

LINEAR PARTIAL DIFFERENCE EQUATIONS
INTRODUCTION
SYMBOLIC METHODS
LAGRANGE’S AND SEPARATION-OF-VARIABLES METHODS
LAPLACE’S METHOD
PARTICULAR SOLUTIONS
SIMULTANEOUS EQUATIONS WITH CONSTANT COEFFICIENTS

NONLINEAR DIFFERENCE EQUATIONS
INTRODUCTION
HOMOGENEOUS EQUATIONS
RICCATI EQUATIONS
CLAIRAUT’S EQUATION
NONLINEAR TRANSFORMATIONS, MISCELLANEOUS FORMS
PARTIAL DIFFERENCE EQUATIONS

APPLICATIONS
INTRODUCTION
MATHEMATICS
PERTURBATION TECHNIQUES
STABILITY OF FIXED POINTS
THE LOGISTIC EQUATION
NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS
PHYSICAL SYSTEMS
ECONOMICS
WARFARE
BIOLOGICAL SCIENCES
SOCIAL SCIENCES
MISCELLANEOUS APPLICATIONS

ADVANCED TOPICS
INTRODUCTION
GENERALIZED METHOD OF SEPARATION OF VARIABLES
CAUCHY–EULER EQUATION
GAMMA AND BETA FUNCTIONS
LAMBERT-W FUNCTION
THE SYMBOLIC CALCULUS
MIXED DIFFERENTIAL AND DIFFERENCE EQUATIONS
EULER POLYNOMIALS
FUNCTIONAL EQUATIONS
FUNCTIONAL EQUATION f(x)2 + g(x)2 = 1
EXACT DISCRETIZATIONS OF DIFFERENTIAL EQUATIONS

ADVANCED APPLICATIONS
FINITE DIFFERENCE SCHEME FOR THE RELUGA x – y – z MODEL
DISCRETE-TIME FRACTIONAL POWER DAMPED OSCILLATOR
EXACT FINITE DIFFERENCE REPRESENTATION OF THE MICHAELIS–MENTON EQUATION
DISCRETE DUFFING EQUATION
DISCRETE HAMILTONIAN SYSTEMS
ASYMPTOTICS OF SCHRODINGER-TYPE DIFFERENCE EQUATIONS
BLACK–SCHOLES EQUATIONS

Appendix: Useful Mathematical Relations

Bibliography

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