降階法及其在偏微分方程數(shù)值解中的應(yīng)用（英文版）

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 fictitious 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 SchrSdinger 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