Difference Equations Theory Applications and Advanced Topics 3rd Edition by Ronald Mickens ISBN 1482230798 9781482230796

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Product details:

ISBN 10: 1482230798

ISBN 13: 9781482230796

Author:  Ronald E. Mickens 

Difference Equations: Theory, Applications and Advanced Topics 3rd Edition: 

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.

 

Difference Equations: Theory, Applications and Advanced Topics 3rd Edition Table of contents:

Part 1: 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

Part 2: First-order difference equations

• Introduction
• General linear equation
• Continued fractions
• A general first-order equation: geometrical methods
• A general first-order equation: expansion techniques

Part 3: Linear difference equations

• Introduction
• Linearly independent functions
• Fundamental theorems for homogeneous equations
• Inhomogeneous equations
• Second-order equations
• Sturm–Liouville difference equations

Part 4: 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

Part 5: Linear partial difference equations

• Introduction
• Symbolic methods
• Lagrange’s and separation-of-variables methods
• Laplace’s method
• Particular solutions
• Simultaneous equations with constant coefficients

Part 6: Nonlinear difference equations

• Introduction
• Homogeneous equations
• Riccati equations
• Clairaut’s equation
• Nonlinear transformations, miscellaneous forms
• Partial difference equations

Part 7: 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

Part 8: 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

Part 9: 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 Schrödinger-type difference equations
• Black–Scholes equations

 

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