Aimin Xu

Some identities involving exponential functions and Stirling numbers and applications

Guo and Qi (2013) posed a problem asking to determine the coefficients a k , i - 1 for 1 ? i ? k such that 1 / ( 1 - e - t ) k = 1 + ? i = 1 k a k , i - 1 ( 1 / ( e t - 1 ) ) ( i - 1 ) . The authors answer this question alternatively by Faa di Brunos formula, unify the eight identities due to Guo and Qi to two identities involving two parameters, and apply them to obtain an explicit expression for the Apostol-Bernoulli numbers and the Fubini numbers, respectively.

A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems

A system of coupled singularly perturbed initial value problems with two small parameters is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solution of the system has boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh a hybrid finite difference scheme is proved to be almost second-order accurate, uniformly in both small parameters. Numerical results supporting the theory are presented.

Finite difference scheme with a moving mesh for pricing Asian options

In this paper we propose a stable numerical method for pricing Asian call options, which is based on a central difference scheme with a moving mesh in the spatial discretization and the Rannacher time stepping scheme in the time discretization. At each time mesh point we make a piecewise uniform mesh to discretise the space interval, which ensures that the matrix associated with the discrete operator is an M-matrix. Hence the spatial discretization scheme is maximum-norm stable for arbitrary volatility and arbitrary interest rate. We show that the scheme is second-order convergent with respect to both time and spatial variables. Numerical results support the theoretical results.

Closed formulas for computing higher-order derivatives of functions involving exponential functions

For integers k ? 1 and n ? 0, the functions 1 / ( 1 - λ e α t ) k and the derivatives ( 1 / ( 1 - λ e α t ) ) ( n ) can be expressed each other by linear combinations. Based on this viewpoint, we find several new closed formulas for higher-order derivatives of trigonometric and hyperbolic functions, derive a higher-order convolution formula for the tangent numbers, and generalize a recurrence relation for the tangent numbers.

A posteriori error analysis for a fractional differential equation

ABSTRACT Numerical treatment for a fractional differential equation (FDE) is proposed and analysed. The solution of the FDE may be singular near certain domain boundaries, which leads to numerical difficulty. We apply the upwind finite difference method to the FDE. The stability properties and a posteriori error analysis for the discrete scheme are given. Then, a posteriori adapted mesh based on a posteriori error analysis is established by equidistributing arc-length monitor function. Numerical experiments illustrate that the upwind finite difference method on a posteriori adapted mesh is more accurate than the method on uniform mesh.

A hybrid finite difference scheme for pricing Asian options

In this paper we apply a hybrid finite difference scheme to evaluate the prices of Asian call options with fixed strike price. We use the Crank-Nicolson method to discretize the time variable and a hybrid finite difference scheme to discretize the spatial variable. The hybrid difference scheme uses the central difference approximation whenever the mesh points are sufficiently away from the left-hand side of the domain to ensure the stability of the scheme; otherwise a midpoint upwind difference scheme is used. The matrix associated with the discrete operator is an M-matrix, which ensures that the spatial discretization scheme is maximum-norm stable. It is proved that the scheme is second-order convergent with respect to both time and spatial variables. Numerical experiments support these theoretical results.

A robust numerical method for a fractional differential equation

Abstract This paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order α  ∈ (0, 1). First the fractional differential equation is transformed into an equivalent Volterra integral equation of the second kind with a weakly singular kernel. Based on the a priori information about the exact solution, an integral discretization scheme on an a priori chosen adapted mesh is proposed. By applying the truncation error estimate techniques and a discrete analogue of Gronwall’s inequality, it is proved that the numerical method is first-order convergent in the discrete maximum norm. Numerical results indicate that this method is more accurate and robust than finite difference methods when α is close to 0.

A robust upwind difference scheme for pricing perpetual American put options under stochastic volatility

In this paper, we present an upwind difference scheme for the valuation of perpetual American put options, using Hestons stochastic volatility model. The matrix associated with the discrete operator is an M-matrix, which ensure that the scheme is stable. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. Numerical results support the theoretical results.

Representation for the Lagrangian numerical differentiation formula involving elementary symmetric functions

By using elementary symmetric functions, this paper presents an explicit representation for the Lagrangian numerical differentiation formula as well as the error estimate for local approximation. And we also point out that the numerical differentiation formula constructed by Li [J.P. Li, General explicit difference formulas for numerical differentiation, J. Comput. Appl. Math. 183 (2005) 29-52] is actually a special case of the Lagrangian numerical differentiation formula to approximate the values of the derivatives at the nodes.

A high-order finite difference scheme for a singularly perturbed fourth-order ordinary differential equation

ABSTRACT In this paper a singularly perturbed fourth-order ordinary differential equation is considered. The differential equation is transformed into a coupled system of singularly perturbed equations. A hybrid finite difference scheme on a Vulanović–Shishkin mesh is used to discretize the system. This hybrid difference scheme is a combination of a non-equidistant generalization of the Numerov scheme and the central difference scheme based on the relation between the local mesh widths and the perturbation parameter. We will show that the scheme is maximum-norm stable, although the difference scheme may not satisfy the maximum principle. The scheme is proved to be almost fourth-order uniformly convergent in the discrete maximum norm. Numerical results are presented for supporting the theoretical results.