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.