An efficient method for solving nonlinear singular initial value problems

In [M.M. Hosseini, Modified Adomain decomposition method for specific second order ordinary differential equations, Appl. Math. Comput. 186 (2007), pp. 117–123] an efficient modification of Adomian decomposition method has been proposed for solving some cases of ordinary differential equations. In this paper, this method is generalized to more cases. The proposed method can be applied to linear, nonlinear, singular and nonsingular problems. Here, it is focused on nonlinear singular initial value problems of ordinary differential equations. The scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method.     

Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems

Cui et al. [M. Cui and F. Geng, Solving singular two point boundary value problems in reproducing kernel space, J. Comput. Appl. Math. 205 (2007), pp. 6–15; H. Yao and M. Cui, A new algorithm for a class of singular boundary value problems, Appl. Math. Comput. 186 (2007), pp. 1183–1191] presents an algorithm to solve a class of singular linear boundary value problems in the reproducing kernel space. In this paper, we will present three new algorithms to solve a class of singular weakly nonlinear boundary value problems in reproducing kernel space. The algorithms are efficiently applied to solving some model problems. It is demonstrated by the numerical examples that those algorithms are highly accurate.     

A new superconvergent method for systems of nonlinear singular boundary value problems

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h ⁶) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h ⁸) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.     

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.     

A novel method for nonlinear singular fourth order four-point boundary value problems

In this paper, a novel method is proposed for solving nonlinear singular fourth order four-point boundary value problems (BVPs) by combining advantages of the homotopy perturbed method (HPM) and the reproducing kernel method (RKM). Some numerical examples are presented to illustrate the strength of the method.     

The iterated defect correction methods for singular two-point boundary value problems

In this paper, the iterated defect correction (IDeC) techniques based on the centered Euler method for the equivalent first order system of the singular two-point boundary value problem in linear case (x ^α y′(x))′ = f(x), y(0) = a,y(1) = b, where 0 < α < 1 are considered. By using the asymptotic expansion of the global error, it is analyzed that the IDeC methods improved the approximate results by means of IDeC steps and the degree of the interpolating polynomials used. Some numerical examples from the literature are given in illustration of this theory.     

Differential quadrature method (DQM) for a class of singular two-point boundary value problems

Differential quadrature method is applied in this work to solve singular two-point boundary value problems with a linear or non-linear nature. It is demonstrated through numerical examples that accurate results for the problem with different types of boundary conditions can be obtained using a considerably small number of grid points. The relative, root mean square and maximum absolute errors in computed solutions are given to show the performance of the method.     

A fixed-point approach for nonlinear discrete boundary value problems

Existence results are established for second-order discrete boundary value problems.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

Automatically identifying the asymptotic behavior of nonlinear singularly perturbed boundary value problems

We describe a computational procedure designed to automatically analyze the behavior of certain general classes of nonlinear singular perturbation problems by applying the combined results of a body of theory that proves the existence of solutions for these problems. We have also created a computer program that implements the computational procedure. The core mathematical knowledge contained in our program is composed of rules that embody the results of mathematical theorems from nonlinear singular perturbation theory. The principle method of proof used in the mathematical theory yields an estimate of a solution by constructing sharp bounding functions that define a region in which the solution exists uniquely. As a result, a successful application of our program produces an approximation of a solution as a side effect. In addition, the mathematical theory can be used to show the existence of multiple solutions for a nonlinear singularly perturbed boundary value problem. This feature is also reflected in the results obtained from our program. The ability to construct such a program depends critically on the successful coupling of a non-deterministic programming technique called path-finding with the capabilities of a computer algebra system. The research reported here was supported in part by the National Science Foundation under NSF Grant EE-14937, and in part by the CAIP Center, Rutgers University, with funds provided by the New Jersey Commission on Science and Technology and by CAIP's industrial members.     

Homotopy perturbation method for solving sixth-order boundary value problems

In this paper, we apply the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations. This equivalent formulation is obtained by using a suitable transformation. The analytical results of the integral equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method. Comparisons are made to confirm the reliability of the homotopy perturbation method. We have also considered an example where the homotopy perturbation method is not reliable.     

Second-order two-point boundary value problems using the variational iteration algorithm-II

In this paper, we introduce an improved variational iteration method (VIM) for nonlinear second-order boundary value problems. The main advantage of this modification is that it can avoid additional computation in determining the unknown parameters in initial approximation when solving boundary value problems using the conventional VIM. Also, iterative sequences obtained using the improved VIM do satisfy the boundary conditions while iterative sequences obtained using conventional VIM may not, in general, satisfy the boundary conditions. Numerical results reveal that the improved method is accurate and efficient.