Chapter 1 The Method of Order Reduction 1.1 Introduction 1.2 First order off-center difference method 1.3 Second order off-center difference method 1.4 Method of fctitious domain 1.5 Method of order reduction 1.6 Comparisons of the four difference methods 1.7 Conclusion Chapter 2 Linear Parabolic Equations 2.1 Introduction 2.2 Derivative boundary conditions 2.3 Derivation of the difference scheme 2.4 A priori estimate for the difference solution 2.5 Solvability,stability and convergence 2.6 Two dimensional parabolic equations 2.7 Conclusion Chapter 3 Linear Hyperbolic Equations 3.1 Introduction 3.2 Derivation of the difference scheme 3.3 A priori estimate 3.4 Solvability,stability and convergence 3.5 Numerical examples 3.6 Conclusion Chapter 4 Linear Elliptic Equations 4.1 Introduction 4.2 Derivation of the difference scheme 4.3 Solvability,stability and convergence 4.4 The Neumann boundary value problem 4.5 A numerical example 4.6 Conclusion Chapter 5 Heat Equations with an Inner Boundary Condition 5.1 Introduction 5.2 Derivation of the difference scheme 5.3 Solvability,stability and convergence 5.4 A numerical example 5.5 Conclusion Chapter 6 Heat Equations with a Nonlinear Boundary Condition 6.1 Introduction 6.2 Derivation of the difference scheme 6.3 Convergence of the difference scheme 6.4 Unique solvability of the difference scheme 6.5 Iterative algorithm and a numerical example 6.6 Conclusion Chapter 7 Nonlocal Parabolic Equations 7.1 Introduction 7.2 Derivation of the difference scheme 7.3 A prior estimate 7.4 Convergence and solvability 7.5 Extrapolation method 7.6 Implementation of the difference scheme 7.7 Conclusion Chapter 8 Fractional Diffusion-wave Equations 8.1 Introduction 8.2 Approximation of the fractional order derivatives 8.3 Derivation of the difference scheme 8.4 Analysis of the difference scheme 8.5 A compact difference scheme 8.6 A slow diffusion system 8.7 A numerical example 8.8 Conclusion Chapter 9 Wave Equations with Heat Conduction 9.1 Introduction 9.2 Boundary conditions 9.3 Derivation of the difference scheme 9.4 Solvability,stability and convergence 9.5 A practical recurrence algorithm 9.6 The degenerate problem 9.7 Conclusion Chapter 10 Timoshenko Beam Equations with Boundary Feedback 10.1 Introduction 10.2 Derivation of the difference scheme 10.3 Analysis of the difference scheme 10.4 A numerical example 10.5 Conclusion Chapter 11 Thermoplastic Problems with Unilateral Constraint 11.1 Introduction 11.2 Derivation of the difference scheme 11.3 Stability and convergence 11.4 Numerical examples 11.5 Conclusion Chapter 12 Thermoelastic Problems with Two-rod Contact 12.1 Introduction 12.2 Derivation of the difference scheme 12.3 Stability and convergence 12.4 Solvability and iterative algorithm 12.5 Numerical examples 12.6 Conclusion Chapter 13 Nonlinear Parabolic Systems 13.1 Introduction 13.2 Difference scheme 13.3 Unique solvability and convergence 13.4 A numerical example 13.5 Conclusion Chapter 14 Heat Equations in Unbounded Domains 14.1 Introduction 14.2 Derivation of the difference scheme 14.3 Analysis of the difference scheme 14.4 A numerical example 14.5 Conclusion Chapter 15 Heat Equations on a Long Strip 15.1 Introduction 15.2 Derivation of the difference scheme 15.3 Analysis of the difference scheme 15.4 A numerical example 15.5 Conclusion Chapter 16 Burgers Equations in Unbounded Domains 16.1 Introduction 16.2 Reformulation of the problem 16.3 Derivation of the difference scheme 16.4 Solvability and stability of the difference scheme 16.5 Convergence of the difference scheme 16.6 A numerical example 16.7 Conclusion Chapter 17 Superthermal Electron Transport Equations 17.1 Introduction 17.2 Derivation of the difference scheme 17.3 Analysis of the difference scheme 17.4 A numerical example 17.5 Conclusion Chapter 18 A Model in Oil Deposit 18.1 Introduction 18.2 Difference scheme and the main results 18.3 Derivation of the difference scheme 18.4 Solvability and convergence 18.5 Conclusion Chapter 19 The Two-dimensional Cahn-Hillard Equation 19.1 Introduction 19.2 Derivation of the difference scheme 19.3 Solvability and convergence of the difference scheme 19.4 Conclusion Chapter 20 ADI and Compact ADI Methods 20.1 Introduction 20.2 Notations and auxiliary lemmas 20.3 Error analysis of the ADI solution and its extrapolation 20.4 Error estimates of the compact ADI method 20.5 A numerical example 20.6 Conclusion Chapter 21 Time-dependent SchrÄodinger Equations 21.1 Introduction 21.2 One-dimensional Crank-Nicolson scheme 21.3 An extension to the high-order compact scheme 21.4 Extensions to multidimensional problems 21.5 Treatment of the nonhomogeneous boundary conditions 21.6 A numerical example 21.7 Conclusion Bibliography
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