An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

Numerical solution of a system of fourth order boundary value problems using variational iteration method

In this paper, the variational iteration method is used to solve a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. Numerical solution obtained by the method is of high accuracy. Moreover, the higher-order derivatives of numerical solution can also approximate the higher-order derivatives of exact solution well. Five examples compared with those considered by Siddiqi and Akram [S.S. Siddiqi, G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Applied Mathematics and Computation 190 (2007) 652–661] show that the method is more efficient.     

Piecewise reproducing kernel method for singularly perturbed delay initial value problems

A direct application of the reproducing kernel method presented in the previous works cannot yield accurate approximate solutions for singularly perturbed delay differential equations. In this letter, we construct a new numerical method called piecewise reproducing kernel method for singularly perturbed delay initial value problems. Numerical results show that the present method does not share the drawback of standard reproducing kernel method and is an effective method for the considered singularly perturbed delay initial value problems.     

A comparison between the variational iteration method and the successive approximations method

In this paper we investigate and compare the variational iteration method and the successive approximations method for solving a class of nonlinear differential equations. We prove that these two methods are equivalent for solving these types of equations.     

He's variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation

In this paper, we apply He''s Variational iteration method (VIM) for solving nonlinear Newell-Whitehead-Segel equation. By using this method three different cases of Newell-Whitehead-Segel equation have been discussed. Comparison of the obtained result with exact solutions shows that the method used is an effective and highly promising method for solving different cases of nonlinear Newell-Whitehead-Segel equation.     

Periodic solution for strongly nonlinear vibration systems by He's variational iteration method

In this paper, we use variational iteration method for strongly nonlinear oscillators. This method is a combination of the traditional variational iteration and variational method. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy. The method can be easily extended to other nonlinear oscillations and hence widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions

In this study, linear and nonlinear partial differential equations with the nonhomogeneous initial conditions are considered. We used Variational iteration method (VIM) and Homotopy perturbation method (HPM) for solving these equations. Both methods are used to obtain analytic solutions for different types of differential equations. Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

The homotopy analysis method for solving higher dimensional initial boundary value problems of variable coefficients

In this article, higher dimensional initial boundary value problems of variable coefficients are solved by means of an analytic technique, namely the Homotopy analysis method (HAM). Comparisons are made between the Adomian decomposition method (ADM), the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics

In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time-fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative ( D _x and D _t), fractional-order derivative ( and ), fractional-order integration ( and ) and delay terms ( and ) of approximated function u(x,t). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time-fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine-Gordon, parabolic integro-differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi-dimensional physical problems of highly nonlinear fractional or variable order of complex nature.     

A Gauss-Seidel iteration method for nonlinear variational inequality problems over rectangles

This paper presents a simple yet practically useful Gauss-Seidel iterative method for solving a class of nonlinear variational inequality problems over rectangles and of nonlinear complementarity problems. This scheme is a nonlinear generalization of a robust iterative method for linear complementarity problems developed by Mangasarian. Global convergence is presented for problems with Z-functions. It is noted that the suggested method can be viewed as a specific case of a class of linear approximation methods studied by Pang and others.     

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

     

Modified defect correction algorithms for ODEs. Part II: Stiff initial value problems

As shown in part I of this paper and references therein, the classical method of Iterated Defect Correction (IDeC) can be modified in several nontrivial ways, extending the flexibility and range of applications of this approach. The essential point is an adequate definition of the defect, resulting in a significantly more robust convergence behavior of the IDeC iteration, in particular, for nonequidistant grids. The present part II is devoted to the efficient high-order integration of stiff initial value problems. By means of model problem investigation and systematic numerical experiments with a set of stiff test problems, our new versions of defect correction are systematically evaluated, and further algorithmic measures are proposed for the stiff case. The performance of the different variants under consideration is compared, and it is shown how strong coupling between non-stiff and stiff components can be successfully handled. AMS subject classification 65L05 Supported by the Austrian Research Fund (FWF) grant P-15030.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Generalized solutions of parabolic initial value problems with discontinuous nonlinearities

We proved recently that parabolic initial value problems with discontinuous nonlinearities have no unique weak solution in general, but have a unique generalized solution in the sense of Colombeau. In this paper we study the relationship between generalized solutions and weak solutions.     

A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

一类含有p-拉普拉斯算子的二阶边值问题的单调迭代正解（英文）

研究了一类含有p-拉普拉斯算子的微分方程积分边值问题.运用迭代技巧,给出了这一类边值问题的单调正解,值得感兴趣的是微分方程中的非线性项含有一阶导数.     

Wavelet analysis method for solving linear and nonlinear singular boundary value problems

In this paper, a robust and accurate algorithm for solving both linear and nonlinear singular boundary value problems is proposed. We introduce the Chebyshev wavelets operational matrix of derivative and product operation matrix. Chebyshev wavelets expansions together with operational matrix of derivative are employed to solve ordinary differential equations in which, at least, one of the coefficient functions or solution function is not analytic. Several examples are included to illustrate the efficiency and accuracy of the proposed method.