Nonlinear numerical analysis in the reproducing Kernel space / by Minggen Cui and Yingzhen Lin.

Format:

Book

Author/Creator:

Cui, Minggen.

Contributor:

Lin, Yingzhen.

Language:

English

Subjects (All):

Hilbert space.

Nonlinear difference equations--Numerical solutions.

Nonlinear difference equations.

Physical Description:

1 online resource (242 p.)

Edition:

1st ed.

Place of Publication:

New York : Nova Science Publishers, c2008.

Language Note:

English

Summary:

Introduces readers engaged in mathematical application the solutions, especially the constructing theory of the reproducing kernel space that the authors originally created and gradually improved.

Contents:

Intro

Nonlinear Numerical Analysisin the Reproducing KernelSpace

Contents

Foreword

Part I

Fundamental Concepts ofReproducing Kernel Space

1.1 Definition of Reproducing Kernel Space

1.2 Fundamental Properties of Reproducing Kernel

1.3 Reproducing Kernel Space Wm2 [a, b] and its ReproducingKernel Function

1.3.1 Absolutely Continuous Function and Some Properties

1.3.2 Function Space Wm2 [a, b] is a Hilbert Space

1.3.3 Function Space Wm2 [a, b] is a Reproducing Kernel Space

1.3.4 Closed Subspaces of the Reproducing Kernel Space Wm2 [a, b]

1.3.5 Two Notes About Reproducing Kernel Space Wm2 [a, b]

1.4 Several Expressions of the Reproducing Kernel ofWm2 [0, 1] or oWm2 [0, 1]

1.5 The Binary Reproducing Kernel Space W(m,n)2 (D)

1.5.1 The Binary Completely Continuous Functions and Some Properties

1.5.2 The Binary Function Space W(m,n)2 (D) is a Hilbert space

1.5.3 The Binary Function Space W(m,n)2 (D) is a Reproducing KernelSpace

1.6 The Reproducing Kernel Space W12 (R)

Some Linear Problems

2.1 Solving Singular Boundary Value Problems

2.1.1 Introduction

2.1.2 The Reproducing Kernel Spaces

2.1.3 Primary Theorem and the Method of Solving Eq. (2.1.1)

2.1.4 The Structure of Solution to Operator Eq. (2.1.3)

2.1.5 Numerical experiments

2.2 Solving the third-order obstacle problems

2.2.1 Introduction

2.2.2 Reproducing Kernel Space oW32 [0, 1]

2.2.3 A bounded linear operator on oW32 [0, 1]

2.2.4 To Solve Eq. (2.2.5)

2.2.5 Numerical Experiments

2.3 Solving Third-Order Singularly Perturbed Problems

2.3.1 Introduction

2.3.2 Asymptotic Expansion Approximation

2.3.3 Several Reproducing Kernel Spaces and Lemmas

2.3.4 The Representation of Solution of TVP (2.3.6)

2.3.5 Numerical Experiments.

2.4 Solving a Class of Variable Delay Integro-DifferentialEquations

2.4.1 Introduction

2.4.2 The Reproducing Kernel Spaces

2.4.3 Linear Operator L on oW22 [0,1)

2.4.4 Two Function Sequences: rn(x), ˆn(x)

2.4.5 The Representation of Solution of Eq. (2.4.4)

2.4.6 Numerical Experiments

Some Algebras Problems

3.1 Solving Infinite System of Linear Equations

3.1.1 Introduction

3.1.2 A Norm-Preserving Operator ˆ from `2 onto W12 [0, 1]

3.1.3 Transform Infinite System of Linear Equation Ay = b intoOperator Equation Ku = f on W12 [0, 1]

3.1.4 Representation of the Solution for Infinite System of LinearEquations Ay = b

3.1.5 Recursion Relation

3.1.6 Numerical Experiments

3.2 A Solution of Infinite System of Quadratic Equations

3.2.1 Introduction

3.2.2 Linear Operators in Reproducing Kernel Space

3.2.3 Separated Solution of (3.2.10)

Part II

Integral equations

4.1 Solving Fredholm Integral Equations of the FirstKind and A Stability Analysis

4.1.1 Introduction

4.1.2 Representation of Exact Solution for Fredholm Integral Equationof the First Kind

4.1.3 The Stability of the Solution on the Eq. (4.1.3)

4.1.4 Numerical Experiments

4.2 Solving Nonlinear Volterra-Fredholm IntegralEquations

4.2.1 Introduction

4.2.2 Theoretic Basis of the Method

4.2.3 Implementations of the Method

4.2.4 Numerical Experiment

4.3 Solving a Class of Nonlinear Volterra-FredholmIntegral Equations

4.3.1 Introduction

4.3.2 Solving Eq. (4.3.1) in the Reproducing Kernel Space

4.3.3 Numerical Experiments

4.4 New Algorithm for Nonlinear Integro-DifferentialEquations

4.4.1 Introduction

4.4.2 Solving the Nonlinear Operator Equation

4.4.3 The Algorithm of Finding the Separable Solution

4.4.4 Numerical Experiments

Differential Equations.

5.1 Solving Variable-Coefficient Burgers Equation

5.1.1 Introduction

5.1.2 The Solution of Eq. (5.1.3)

5.1.3 The Implementation Method

5.1.4 Numerical Experiments

5.2 The Nonlinear Age-Structured Population Model

5.2.1 Numerical Experiments

5.2.2 Solving Population Model can be Turned into Solving OperatorEquation (IV)

5.2.3-1 Solving Eq. (II) can turned into solving Eq. (IV)

5.2.3-2 The Boundedness of Operators

5.2.3 The Exact Solution of Eq.(IV)

5.2.4-1 Solving Eq. (5.2.31) can be Turned into Finding the SeparableSolution of Eq. (5.2.34)

5.2.4-2 The Analytic Representation of all Solutions of Lu = f

5.2.4-3 The Representation of the Exact Solution of Eq. (5.2.31)

5.2.4-4 The Numerical Algorithm for Solving the " ApproximatelySolution of Eq. (5.2.31)

5.2.4 Numerical Experiments

5.3 Solving a Kind of Nonlinear Partial DifferentialEquations

5.3.1 Introduction

5.3.2 Transformation of the Nonlinear Partial Differential Equation

5.3.3 The Definition of Operator L

5.3.4 Decomposition into Direct Sum of oW(2,3)2 (D)

5.3.5 Solving the Nonlinear Partial Differential Equation

5.3.6 Numerical Experiments

5.4 Solving the Damped Nonlinear Klein-GordonEquation

5.4.1 Introduction

5.4.2 Linear Operator on Reproducing Kernel Spaces

5.4.3 The Solution of Eq. (5.4.3)

5.4.4 Numerical experiments

5.4.5 Conclusion

5.5 Solving a Nonlinear Second Order System

5.5.1 Introduction

5.5.2 Several Reproducing Kernel Spaces and Lemmas

5.5.3 The Analytical and Approximate Solutions of Eq. (5.5.2)

5.5.3-1 The Implementation Method

5.5.4 Numerical Experiments

5.6 To Solve a Class of Nonlinear Differential Equations

5.6.1 Introduction

5.6.2 Linear Operator on Reproducing Kernel Spaces

5.6.3 Direct Sum of oW(3,1)2 (D)

5.6.4 Solution of (Lw)(x) = f(x)

5.6.5 Example.

The Exact Solution of NonlinearOperator Equation

6.1 Introduction

6.1.1 Preliminary Knowledge

6.1.2 Operator K

6.1.3 About Eq. (6.1.10) and Eq. (6.1.6)

6.1.4 Solving Eq. (6.1.10)

6.1.5 Numerical Experiments

6.2 All Solutions of System of Ill-Posed OperatorEquations of the First Kind

6.2.1 Introduction

6.2.2 Lemmas

6.2.3 Solving Au = f in Reproducing Kernel Sapce

6.2.4 Numerical Experiments

Solving the Inverse Problems

7.1 Solving the Coefficient Inverse Problem

7.1.1 Introduction

7.1.2 The Reproducing Kernel Spaces

7.1.3 Transformation of Eq. (7.1.1)

7.1.4 Decomposition into Direct Sum of oW(3,3)2 (D)

7.1.5 The Method of Solving Eq. (7.1.6)

7.1.6 Numerical Experiments

7.2 A Determination of an Unknown Parameterin Parabolic Equations

7.2.1 Introduction

7.2.2 The Exact Solution of Eq. (7.2.4)

7.2.3 An Iteration Procedure

7.2.4 Numerical Experiments

Bibliography

INDEX.

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 207-222) and index.

Description based on print version record.

ISBN:

1-61470-436-8

OCLC:

923661080