Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach

A number of new layer methods for solving the Dirichlet problemfor semilinear parabolic equations are constructed by usingprobabilistic representations of their solutions. The methodsexploit the ideas of weak sense numerical integration of stochasticdifferential equations in a bounded domain. Despite their probabilisticnature these methods are nevertheless deterministic. Some convergencetheorems are proved. Numerical tests are presented.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

Boundary and initial-value methods for solving Fredholm equations with semidegenerate kernels

We develop a new approach to the theory and numerical solution of a class of linear and nonlinear Fredholm equations. These equations, which have semidegenerate kernels, are shown to be equivalent to two-point boundary-value problems for a system of ordinary differential equations. Applications of numerical methods for this class of problems allows us to develop a new class of numerical algorithms for the original integral equation. The scope of the paper is primarily theoretical; developing the necessary Fredholm theory and giving comparisons with related methods. For convolution equations, the theory is related to that of boundary-value problems in an appropriate Hilbert space. We believe that the results here have independent interest. In the last section, our methods are extended to certain classes of integrodifferential equations.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

Stein implicit Runge-Kutta methods with high stage order for large-scale ordinary differential equations

The Runge-Kutta method is one of the most popular implicit methods for the solution of stiff ordinary differential equations. For large problems, the main drawback of such methods is the cost required at each integration step for computing the solution of a nonlinear system of equations. In this paper, we propose to reduce the cost of the computation by transforming the linear systems arising in the application of Newton's method to Stein matrix equations. We propose an iterative projection method onto block Krylov subspaces for solving numerically such Stein matrix equations. Numerical examples are given to illustrate the performance of our proposed method.     

Numerical solutions of integral and integro-differential equations using Legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.      

On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.     

Collocation methods for solving multivariable integral equations of the second kind

In a recent paper (Allouch, in press) [5] on one dimensional integral equations of the second kind, we have introduced new collocation methods. These methods are based on an interpolatory projection at Gauss points onto a space of discontinuous piecewise polynomials of degree r$r$ which are inspired by Kulkarni’s methods (Kulkarni, 2003) [10], and have been shown to give a 4r+4$4r+4$ convergence for suitable smooth kernels. In this paper, these methods are extended to multi-dimensional second kind equations and are shown to have a convergence of order 2r+4$2r+4$. The size of the systems of equations that must be solved in implementing these methods remains the same as for Kulkarni’s methods. A two-grid iteration convergent method for solving the system of equations based on these new methods is also defined.     

时滞积分微分方程的Rosenbrock方法

本文研究时滞积分微分方程的数值方法.通过改造现有常及离散型延迟微分方程的数值方法,并匹配以适当数值求积公式,构造了求解时滞积分微分方程的Rosenbrock方法,导出了其稳定性准则.数值例子阐明了所获方法的计算有效性.