3rd Edition

Difference Equations Theory, Applications and Advanced Topics, Third Edition

By Ronald E. Mickens Copyright 2015
    555 Pages
    by Chapman & Hall

    555 Pages 46 B/W Illustrations
    by Chapman & Hall

    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.

    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

    Index

    Biography

    Ronald E. Mickens