A numerical scheme for solving nonhomogeneous time-dependent problems

A numerical technique for solving time-dependent problems with variable coefficient governed by the heat, convection diffusion, wave, beam and telegraph equations is presented. The Sinc–Galerkin method is applied to construct the numerical solution. The method is tested on three problems and comparisons are made with the exact solutions. The numerical results demonstrate the reliability and efficiency of using the Sinc–Galerkin method to solve such problems. Received: January 18, 2005     

Iterative splitting methods for solving time-dependent problems

There is increasing motivation for solving time-dependent differential equations with iterative splitting schemes. While Magnus expansion has been intensively studied and widely applied for solving explicitly time-dependent problems, the combination with iterative splitting schemes can open up new areas. The main problems with the Magnus expansion are the exponential character and the difficulty of deriving practical higher order algorithms. An alternative method is based on iterative splitting methods that take into account a temporally inhomogeneous equation. In this work, we show that the ideas derived from the iterative splitting methods can be used to solve time-dependent problems. Examples are discussed.     

A novel numerical scheme for solving Burgers’ equation

In this paper, we propose a novel numerical scheme for solving Burgers’ equation. The scheme is based on a cubic spline quasi-interpolant and multi-node higher order expansion, which make the algorithm simple and easy to implement. The numerical experiments show that the proposed method produces high accurate results.     

A step-by-step method with Laguerre functions for solving time-dependent problems

A step-by-step modification of the well-known approach proposed by Mikhaylenko and Konyukh to solving dynamic problems is proposed. The approach is based on the Laguerre transform with respect to time. In this modification the Laguerre transform is applied to a sequence of finite time intervals. The solution obtained at the end of a time interval is used as the initial data for solving the problem on the next time interval. The method is illustrated by examples for the harmonic oscillator problem and the 1D wave equation. Accuracy and stability of the method are analyzed. This approach allows obtaining a solution of high accuracy on large time intervals.     

A numerical technique for solving fractional variational problems

This paper presents an accurate numerical method for solving a class of fractional variational problems (FVPs). The fractional derivative in these problems is in the Caputo sense. The proposed method is called fractional Chebyshev finite difference method. In this technique, we approximate FVPs and end up with a finite‐dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. The fractional derivative approximation using Clenshaw and Curtis formula introduced here, along with Clenshaw and Curtis procedure for the numerical integration of a non‐singular functions and the Rayleigh–Ritz method for the constrained extremum, is considered. By this method, the given problem is reduced to the problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FVPs. Special attention is given to study the convergence analysis and evaluate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. A comparison with another method is given. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical method for solving nonlinear ill-posed problems

A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered.     

A parameter-uniform implicit difference scheme for solving time-dependent Burgers’ equations

A numerical study is made for solving one dimensional time dependent Burgers’ equation with small coefficient of viscosity. Burgers’ equation is one of the fundamental model equations in the fluid dynamics to describe the shock waves and traffic flows. For high coefficient of viscosity a number of solution methodology exist in the literature [6], [7], [8] and [9] and [14] but for the sufficiently low coefficient of viscosity, the exist solution methodology fail and a discrepancy occurs in the literature. In this paper, we present a numerical method based on finite difference which works nicely for both the cases, i.e., low as well as high viscosity coefficient. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on uniform mesh and a standard upwind finite difference scheme to discretize in spacial direction on piecewise uniform mesh. The quasilinearzation process is used to tackle the non-linearity. An extensive amount of analysis has been carried out to obtain the parameter uniform error estimates which show that the resulting method is uniformly convergent with respect to the parameter. To illustrate the method, numerical examples are solved using the presented method and compare with exact solution for high value of coefficient of viscosity.     

A numerical method for solving singular boundary value problems

Summary A numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity. A regular boundary value problem is derived on the remaining interval, for which a difference method is used. Convergence theorems are given for general schemes and for schemes of positive type for second order equations.     

非光滑最优控制问题的一种数值解法

针对非光滑最优控制问题提出一种分段数值解法.首先对问题进行全局拟谱离散,然后选取分点,将时间区域进行剖分,在每段区域上对问题进行离散,离散过程采用Chebyshev-Legendre拟谱方法,可以有效借助快速Legendre变换提高算法的运算效率,比现有算法在很大程度上节省了计算时间.给出了相关的理论分析,数值结果表明方法的高精度和有效性.     

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

解对流占优反应扩散问题一致稳定的差分格式

通过将一般的反应扩散方程转化为主部为守恒型方程形式,构造出一种稳定和高精度的新型差分格式.这种差分格式最大的优点是具有与方程Peclet数和网格步长无关的一致稳定性,特别适合求解强对流占优问题或边界层问题.同时还给出了差分格式按L_∞模的一致稳定性和O(h~2)阶收敛速度的理论分析.数值实验验证了理论分析结果.     

A brief survey on numerical methods for solving singularly perturbed problems

In the present paper, a brief survey on computational techniques for the different classes of singularly perturbed problems is given. This survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000-2009. However some older important relevant papers are also included in this survey. We also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223-259, 130 (2002) 457-510, 134 (2003) 371-429] for details. Thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.     

A numerical technique for solving a class of fractional variational problems

This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.     

An efficient discretization scheme for solving ill-posed problems

In this paper, we consider a finite-dimensional approximation scheme combined with Tikhonov regularization for solving ill-posed problems. Error estimates are obtained by an a priori parameter choice strategy and the results show that the amount of discrete information required for solving the problem is far less than the traditional finite-dimensional approach.     

Some numerical techniques for solving elliptic interface problems

Many physical phenomena can be modeled by partial differential equations with singularities and interfaces. The standard finite difference and finite element methods may not be successful in giving satisfactory numerical results for such problems. Hence, many new methods have been developed. Some of them are developed with the modifications in the standard methods, so that they can deal with the discontinuities and the singularities. In this article, a survey has been done on some recent efficient techniques to solve elliptic interface problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 94‐114, 2012     

A numerical method for solving optimal control problems using state parametrization

A numerical method for solving a special class of optimal control problems is given. The solution is based on state parametrization as a polynomial with unknown coefficients. This converts the problem to a non-linear optimization problem. To facilitate the computation of optimal coefficients, an improved iterative method is suggested. Convergence of this iterative method and its implementation for numerical examples are also given.     

A numerical method for solving boundary value problems for fractional differential equations

A numerical scheme, based on the Haar wavelet operational matrices of integration for solving linear two-point and multi-point boundary value problems for fractional differential equations is presented. The operational matrices are utilized to reduce the fractional differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.     

Synthesis scheme for hybrid methods of solving boundary problems

A scheme is proposed for the development of a numericoanalytic boundary problem technique that can synthesize the mesh-projective method with the integral transform method to yield an effective hybrid method for the determination of the temperature field in a body from a moving heat source with given dependences of the thermophysical parameters on temperature.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 19–22, 1987.