Partial Difference Equations (Advances in Discrete Mathematics and Applications, 3)

Partial Difference Equations (Advances in Discrete Mathematics and Applications, 3)

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  • 製本 Hardcover:ハードカバー版/ページ数 272 p.
  • 言語 ENG,ENG
  • 商品コード 9780415298841
  • DDC分類 515.625

基本説明

Applications in signal processing, image processing, numerical simulations, and population dynamics. A concise introduction.

Full Description


Partial Difference Equations treats this major class of functional relations. Such equations have recursive structures so that the usual concepts of increments are important. This book describes mathematical methods that help in dealing with recurrence relations that govern the behavior of variables such as population size and stock price. It is helpful for anyone who has mastered undergraduate mathematical concepts. It offers a concise introduction to the tools and techniques that have proven successful in obtaining results in partial difference equations.

Table of Contents

Series Editors' Preface                            ix
Preface xi
1 Modelling 1 (16)
1.1 Introduction 1 (1)
1.2 Examples 2 (12)
1.2.1 Discrete Heat Equations 2 (2)
1.2.2 Two-Level Equations 4 (6)
1.2.3 Multi-Level Equations 10 (1)
1.2.4 Implicit Reaction Diffusion 11 (1)
Equations
1.2.5 Discrete Time Independent Equations 12 (2)
1.3 Auxiliary Conditions 14 (2)
1.4 Notes and Remarks 16 (1)
2 Basic Tools 17 (30)
2.1 Subsets of the Lattice Plane 17 (2)
2.2 Classifications of Partial Difference 19 (2)
Equations
2.3 Finite Differences 21 (4)
2.4 Summable Infinite sequences 25 (6)
2.5 Convolution of Doubly Infinite Sequences 31 (2)
2.6 Frequency Measures 33 (7)
2.7 Useful Results For Matrices 40 (3)
2.8 Discrete Gronwall Inequalities 43 (1)
2.9 Miscellaneous 44 (1)
2.10 Notes and Remarks 45 (2)
3 Symbolic Calculus 47 (46)
3.1 Introduction 47 (1)
3.2 Semi-infinite Univariate Sequences 47 (27)
3.2.1 Ring of sequences 47 (3)
3.2.2 Operators 50 (1)
3.2.3 Summation Operators 51 (3)
3.2.4 Translation or Shift Operators 54 (3)
3.2.5 Rational Operators 57 (2)
3.2.6 Attenuation Operators 59 (1)
3.2.7 Sequences and series of Operators 60 (4)
3.2.8 Algebraic Derivatives 64 (5)
3.2.9 Algebraic Integrals 69 (2)
3.2.10 Ordinary Difference Equations 71 (3)
3.3 Semi-Infinite Bivariate Sequences 74 (18)
3.3.1 Ring of Double Sequences 74 (6)
3.3.2 Operators 80 (1)
3.3.3 Separable Double Sequences 81 (3)
3.3.4 Basic Relations Between Operators 84 (1)
3.3.5 Attenuation Operators 85 (1)
3.3.6 Sequences and Series of Operators 86 (4)
3.3.7 Algebraic Derivatives 90 (2)
3.4 Notes and Remarks 92 (1)
4 Monotonicity and Convexity 93 (20)
4.1 Introduction 93 (1)
4.2 Univariate Maximum Principles 93 (3)
4.3 Bivariate Maximum Principles 96 (4)
4.4 Univariate Wirtinger's Inequalities 100(3)
4.5 Bivariate Wirtinger's Inequalities 103(8)
4.6 Notes and Remarks 111(2)
5 Explicit solutions 113(34)
5.1 Introduction 113(4)
5.2 Formal Methods 117(3)
5.3 The Method of Translation 120(1)
5.4 The Method of Operators 121(4)
5.5 The Method of Separable Solut卲ns 125(3)
5.6 The Method of Convolution 128(13)
5.6.1 Two-Level Equations over the Upper 128(6)
Half Lattice Plane
5.6.2 Three-Level Equations over the 134(7)
Upper Half Lattice Plane
5.7 Method of Linear Systems 141(4)
5.8 Notes and Remarks 145(2)
6 Stability 147(54)
6.1 Stability Concepts 147(1)
6.2 Equations Over Cylinders 148(13)
6.2.1 Method of General Solutions 148(1)
6.2.2 Method of Maximum Principles 149(1)
6.2.3 Method of Energies 150(1)
6.2.4 Method of Functional Inequalities 151(3)
6.2.5 Spectral Methods 154(5)
6.2.6 Method of Separable Solutions 159(2)
6.3 Equations Over Half Planes 161(12)
6.3.1 Method of Exact Solutions for 162(2)
Two-Level Equations
6.3.2 Method of Exact Solutions for 164(5)
Three-Level Equations
6.3.3 Method of Induction for Three-Level 169(4)
Equations
6.4 Equations Over Quadrants 173(27)
6.4.1 Method of Exact Solutions for a 174(1)
Two-Level Equation
6.4.2 Method of Induction for a Two-Level 175(6)
Nonhomogeneous Equation
6.4.3 Method of Induction for a 181(7)
Four-Point Equation
6.4.4 Method of Induction for a 188(4)
Four-Point Delay Equation
6.4.5 Method of Induction for a 192(5)
Five-Point Delay Equation
6.5 Equations Over Finite Domains 197(3)
6.6 Notes and Remarks 200(1)
7 Existence 201(22)
7.1 Introduction 201(1)
7.2 Traveling Waves 201(4)
7.3 Positive and Bounded solutions 205(2)
7.4 Monotone Method for a Finite Laplace 207(4)
Equation
7.5 Contraction Method for a Finite Laplace 211(1)
Equation
7.6 Monotone Method for Evolutionary 212(3)
Equations
7.7 Eigenvalue Method for a Boundary Problem 215(1)
7.8 Contraction Method for a Boundary 216(1)
Problem
7.9 Monotone Method for Boundary Problems 217(5)
7.10 Notes and Remarks 222(1)
8 Nonexistence 223(32)
8.1 Introduction 223(1)
8.2 Equations Over The Plane 223(4)
8.3 Equations Over Quadrants 227(8)
8.3.1 Three-Point Equations with Two 227(1)
Constant Coefficients
8.3.2 Four-Point Equations with Three 228(2)
Constant Coefficients
8.3.3 Characteristic Initial Value 230(1)
Problems
8.3.4 Delay Partial Difference Equations 231(3)
8.3.5 Frequently Positive Solutions 234(1)
8.4 Equations Over Cylinders 235(15)
8.4.1 Linear Discrete Heat Equation With 235(3)
Constant Coefficients
8.4.2 Parabolic Type Equations with 238(3)
Variable Coefficients
8.4.3 Discrete Elliptic Equations 241(3)
8.4.4 Initial Boundary Value Problems 244(1)
8.4.5 Linear Hybrid Five-Point Equations 245(5)
8.5 Equations Over Finite Domains 250(2)
8.6 Notes and Remarks 252(3)
Bibliography 255(11)
Index 266