SOLVING NONLINEAR DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH SINGULAR PERTURBATION VIA BLOCK BOUNDARY VALUE METHODS

Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.

Fourth Order Compact Boundary Value Method for Option Pricing with Jumps

We consider pricing options in a jump-diffusion model which requires solving a partial integro-differential equation.Discretizing the spatial direction with a fourth order compact scheme leads to a linear system of ordinary differential equations.For the temporal direction,we utilize the favorable boundary value methods owing to their advantageous stability properties.In addition,the resulting large sparse system can be solved rapidly by the GMRES method with a circulant Strang-type preconditioner.Numerical results demonstrate the high order accuracy of our scheme and the efficiency of the preconditioned GMRES method.

ESSENTIALLY SYMPLECTIC BOUNDARY VALUE METHODS FOR LINEAR HAMILTONIAN SYSTEMS

In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called symplectic intoprators. Several symplectic methods are known in the class of Runge-Kutta methods. However, no higll order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are 'essentially' maintained by the discrete map, at least for linear problems.

BOUNDARY VALUE METHODS FOR CA PUTO FR ACTIONAL DIFFERENTIAL EQUATIONS

This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of extended BVMs for the CFDEs with y-order(0<r<1)Caputo derivatives are derived.The local stability,unique solvability and convergence of the methods are studied.It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at min{p,m-γ+1}.In the end,by performing several numerical examples,the computational efficiency,accuracy and comparability of the methods are further ilustrated.

A New Circulant Preconditioned GMRES Method for Solving Ordinary Differential Equation

The preconditioned generalized minimal residual(GMRES) method is a common method for solving non-symmetric,large and sparse linear systems which originated in discrete ordinary differential equations by Boundary value methods.In this paper,we propose a new circulant preconditioner to speed up the convergence rate of the GMRES method, which is a convex linear combination of P-circulant and Strang-type circulant preconditioners. Theoretical and practical arguments are given to show that this preconditioner is feasible and effective in some cases.

ARBITRARILY HIGH-ORDER ENERGY-CONSERVING METHODS FOR HAMILTONIAN PROBLEMS WITH QUADRATIC HOLONOMIC CONSTRAINTS

In this paper,we define arbitrarily high-order energy-conserving methods for Hamilto-nian systems with quadratic holonomic constraints.The derivation of the methods is made within the so-called line integral framework.Numerical tests to illustrate the theoretical findings are presented.

A DUAL COUPLED METHOD FOR BOUNDARY VALUE PROBLEMS OF PDE WITH COEFFICIENTS OF SMALL PERIOD

In this paper the homogenization method is improved to develop one kind of dual coupled approximate method, which reflects both the macro-scope properties of whole structure and its loadings, and micro-scope configuration properties of composite materials. The boundary value problem of woven membrane is considered, the dual asymptotic expression of the exact solution is given, and its approximation and error estimation are discussed. Finally the numerical example shows the effectiveness of this dual coupled method.

SPECTRAL METHOD FOR MIXED INHOMOGENEOUS BOUNDARY VALUE PROBLEMS IN THREE DIMENSIONS

In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approxima- tion in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inho- mogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

Optimal L_∞ Estimates for Galerkin Methods for Nonlinear Singular Two-point Boundary Value Problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

INITIAL VALUE PROBLEMS AND FIRST BOUNDARY PROBLEMS FOR A CLASS OF QUASILINEAR WAVE EQUATIONS

The initial value problems and the first boundary problems for the quasilinear wave equation u_(tt)-[a_0+na_1(u_x)^(n-1)]u_(xx)-a_2u_(xxtt)=0 are considered,where a_0,a_2>0 are constants,a_1 is an arbitrary real number,n is a natural number.The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

In this paper,an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods.In order to efficiently solve the generated linear large-scale system,the generalized minimal residual(GMRES)algorithm is applied.For accelerating the convergence rate of the it erative,the St rang-type,Chantype and P-type preconditioners are introduced.The suggested met hod can reach higher order accuracy both in space and in time than the existing met hods.When the used boundary value method is Ak1,K2-stable,it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1.It implies that the iterative solution is convergent rapidly.Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.

A Numerical Algorithm for Determination of a Control Parameter in Two-dimensional Parabolic Inverse Problems

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. In this study we present a high order scheme for determining unknown control parameter and unknown solution of two-dimensional parabolic inverse problem with overspe- cialization at a point in the spatial domain. In this approach, a compact fourth-order scheme is used to discretize spatial derivatives of equation and reduces the problem to a system of ordinary differential equations （ODEs）. Then we apply a fourth order boundary value method to the solution of resulting system of ODEs. So the proposed method has fourth order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes in the literature. Also we will investigate the effect of noise in data on the approximate solutions.