Zhi-zhong Sun

A compact finite difference scheme for the fractional sub-diffusion equations

In this paper, a compact finite difference scheme for the fractional sub-diffusion equations is derived. After a transformation of the original problem, the L1 discretization is applied for the time-fractional part and fourth-order accuracy compact approximation for the second-order space derivative. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis. The technique is quite novel and different from previous analytical methods. Finally, a numerical example is provided to show the effectiveness and accuracy of the method.

Interest Rate Modeling

As pointed out in Sect. 2.3, when the short-term interest rate is considered as a random variable, there is an unknown function λ(r, t), called the market price of risk, in the governing equation.

A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications

In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of order α ( 0 < α < 1 ) is developed. It is established by means of the quadratic interpolation approximation using three points ( t j - 2 , f ( t j - 2 ) ) , ( t j - 1 , f ( t j - 1 ) ) and ( t j , f ( t j ) ) for the integrand f ( t ) on each small interval t j - 1 , t j ] ( j ? 2 ), while the linear interpolation approximation is applied on the first small interval t 0 , t 1 ] . As a result, the new formula can be formally viewed as a modification of the classical L1 formula, which is obtained by the piecewise linear approximation for f ( t ) . Both the computational efficiency and numerical accuracy of the new formula are superior to that of the L1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of L1-2 formula. Second, by the new formula, two improved finite difference schemes with high order accuracy in time for solving the time-fractional sub-diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed, respectively. In addition, the application of the new formula into solving fractional ordinary differential equations is also presented. Several numerical examples are computed. The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L1 formula when solving time-fractional differential equations numerically.

A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions

Combining order reduction approach and L1 discretization, a box-type scheme is presented for solving a class of fractional sub-diffusion equation with Neumann boundary conditions. A new inner product and corresponding norm with a Sobolev embedding inequality are introduced. A novel technique is applied in the proof of both stability and convergence. The global convergence order in maximum norm is O(@t^2^-^@a+h^2). The accuracy and efficiency of the scheme are checked by two numerical tests.

A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation

The Cahn-Hilliard equation is a nonlinear evolutionary equation that is of fourth order in space. In this paper a linearized finite difference scheme is derived by the method of reduction of order. It is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. The coefficient matrix of the difference system is symmetric and positive definite, so many well-known iterative methods (e.g. Gauss-Seidel, SOR) can be used to solve the system.

Compact Alternating Direction Implicit Scheme for the Two-Dimensional Fractional Diffusion-Wave Equation

In this paper, we consider the numerical method for solving the two-dimensional fractional diffusion-wave equation with a time fractional derivative of order

A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation

\alpha

Finite difference methods for the time fractional diffusion equation on non-uniform meshes

(

Error Estimates of Crank-Nicolson-Type Difference Schemes for the Subdiffusion Equation

1<\alpha<2

Second-order approximations for variable order fractional derivatives

). A difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping is proposed and analyzed. The unconditional stability and