A modification of classical third order methods is proposed. The main advantage of these methods is that they do not need evaluate any second order Fréchet derivative. A convergence theorem in Banach spaces is analyzed. Finally, some preliminary numerical results are presented. 

Typical solution methods for solving mixed complementarity problems either generate feasible iterates but have to solve relatively complicated subproblems such as quadratic programs or linear complementarity problems, or (those methods) have relatively simple subproblems such as system of linear equations but possibly generate infeasible iterates. In this paper, we propose a new Newton-type method for solving monotone mixed complementarity problems, which ensures to generate feasible iterates, and only has to solve a system of well-conditioned linear equations with reduced dimension per iteration. Without any regularity assumption, we prove that the whole sequence of iterates converges to a solution of the problem (truly globally convergent). Furthermore, under suitable conditions, the local superlinear rate of convergence is also established.

Implicit Runge-Kutta method is highly accurate and stable for stiff initial value problem. But the iteration technique used to solve implicit Runge-Kutta method requires lots of computational efforts. In this paper, we extend the Parallel Diagonal Iterated Runge-Kutta (PDIRK) methods to delay differential equations (DDEs). We give the convergence region of PDIRK methods, and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector method. Finally, we analysis the speed-up factor through a numerical experiment. The results show that the PDIRK methods to DDEs are efficient.

This paper is concerned with numerical methods for American option pricing. We employ numerical analysis and the notion of viscosity solution to show uniform convergence of the explicit difference scheme and the binomial tree method. We also prove the existence and convergence of the optimal exercise boundaries in the above approximations.

This paper deals with the asymptotic stability analysis of $\theta$ – methods for multi-pantograph delay differential equation 

Here $\lambda, μ_1,μ_2, ... , μ_l, u_0 \in C$.

In recent years stability properties of numerical methods for this kind of equation has been studied by numerous authors. Many papers are concerned with meshes with fixed stepsize. In general the developed techniques give rise to non-ordinary recurrence relation. In this work, instead,we study constrained variable stpesize schemes, suggested by theoretical and computational reasons, which lead to a non-stationary difference equation. A general theorem is presented which can be used to obtain the characterization of the stability regions of $\theta$ – methods.

In this paper, we consider an inverse scattering problem for an obstacle $D ⊂ R^2$ with Robin boundary condition. By applying the point source, we give a regularizing method to recover the scattered field from the far-field pattern. Numerical implementations are also presented.  

In this paper a summation formula for finding the surface area of general ellipsoids is derived.

We consider the numerical approximations of the complex amplitude in a coupled bay-river system in this work. One half-circumference is introduced as the artificial boundary in the open sea, and one segment is introduced as the artificial boundary in the river if the river is semi-infinite. On the artificial boundary a sequence of high-order artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.

It is known that for a given matrix $A$ of rank $r$, and a set $D$ of positive diagonal matrices, $\sup_{W\in D}||(W^{\frac{1}{2}}A)^†W^{\frac{1}{2}}||_2=(\min_i \sigma_+(A^{(i)})^{-1}$, in which $(A^{(i)})$is a submatrix of A formed with $r = (\rm{rank}(A))$ rows of $A$, such that $(A^{(i)})$ has full row rank $r$. In many practical applications this value is too large to be used. 

In this paper we consider the case that both $A$ and $W(\in D)$ are fixed with $W$ severely stiff. We show that in this case the weighted pseudoinverse $W^{\frac{1}{2}}A)^†W^{\frac{1}{2}}$ is close to a multi-level constrained weighted pseudoinverse therefore $||(W^{\frac{1}{2}}A)^†W^{\frac{1}{2}}||_2$ is uniformly bounded. We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.

In this paper we present a dynamic optimal method for adjusting the centering parameter in the wide-neighborhood primal-dual interior-point algorithms for linear programming, while the centering parameter is generally a constant in the classical wide-neighborhood primal-dual interior-point algorithms. The computational results show that the new method is more efficient.

In this paper, we deal with the boundedness and the asymptotic stability of linear and one-leg multistep methods for generalized pantograph equations of neutral type, which arise from some fields of engineering. Some criteria of the boundedness and the asymptotic stability for the methods are obtained.

A mutually orthogonal system of rational functions on the whole line is introduced. Some approximation results are established. As an example of applications, a modified Legendre rational spectral scheme is given for the Dirac equation. Its numerical solution keeps the same conservation as the genuine solution. This feature not only leads to reasonable numerical simulation of nonlinear waves, but also simplifies the analysis. The convergence of the proposed scheme is proved. Numerical results demonstrate the efficiency of this new approach and coincide with the analysis well.

In this paper, we construct and analyse a mortar finite volume method for the discretization for the biharmonic problem in $R^2$. This method is based on the mortar-type Adini nonconforming finite element spaces. The optimal order $H^2$-seminorm error estimate between the exact solution and the mortar Adini finite volume solution of the biharmonic equation is established.