The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations

The layout of this book is as follows. Chapter 1 provides a microcosm of the method of order reduction via a two-point boundary value problem. Chapters 2, 3 and 4 are devoted, respectively, to the numerical solutions of linear parabolic, hyperbolic and elliptic equations by the method of order reduction. They are the core of the book. Chapters 5, 6 and 7 respectively consider the numerical approaches to the heat equation with an inner boundary condition, the heat equation with a nonlinear boundary condition and the nonlocal parabolic equation. Chapter 8 discusses the numerical approximation to a fractional diffusion-wave equation. The next five chapters are devoted to the numerical solutions of several coupled systems of differential equations. The numerical procedures for the heat equation and the Burgers equation in the unbounded domains are studied in Chapters 14, 15 and 16. Chapter 17 provides a numerical method for the superthermal electron transport equation, which is a degenerate and nonlocal evolutionary equation. The numerical solution to a model in oil deposit on a moving boundary is presented in Chapter 18. Chapter 19 deals with the numerical solution to the Cahn-Hilliard equation, which is a fourth order nonlinear evolutionary equation. The ADI and compact ADI methods for the multidimensional parabolic problems are discussed in Chapter 20. The numerical errors in the maximum norm are obtained. Chapter 21, the last chapter, is devoted to the numerical solution to the time-dependent SchrSdinger equation in quantum mechanics. ...

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
　……
Chapter 4 Linear Elliptic Equations
Chapter 5 Heat Equations with an Inner Boundary Condition
Chapter 6 Heat Equations with a Nonlinear Boundary Condition
Chapter 7 Nonlocal Parabolic Equations
Chapter 8 Fractional Diffusion-wave Equations
Chapter 9 Wave Equations with Heat Conduction
Chapter 10 Timoshenko Beam Equations with Boundary Feedback
Chapter 11 Thermoplastic Problems with Unilateral Constraint
Chapter 12 Thermoelastic Problems with Two-rod Contact
Chapter 13 Nonlinear Parabolic Systems
Chapter 14 Heat Equations in Unbounded Domains
Chapter 15 Heat Equations on a long Strip
Chapter 16 Burgers Equations in Unbounded Domains
Chapter 17 Superthermal Electron Transport Equations
Chapter 18 A Model in Oil Deposit
Chapter 19 The Two-dimensional Cahn-Hillard Equation
Chapter 20 ADI and Compact ADI Methods
Chapter 21 Time-dependent Schrodinger Equations
Bibliography