Homotopy perturbation technique for solving two-point boundary value problems - comparison with other methods

The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.     

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.     

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.     

Alternating group explicit iterative method for nonlinear singular Fredholm Integro-differential boundary value problems

We discuss alternating group explicit (AGE) iterative method for the numerical solution of linear and non-linear singular integro-differential boundary value problems. The proposed AGE iterative method shows the superiority over the corresponding successive over relaxation (SOR) iterative methods. The presented algorithm is suitable for the use on parallel computers. The convergence analysis is briefly discussed. Computational results are presented to illustrate the utility of the proposed method.     

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.     

A fixed-point approach for nonlinear discrete boundary value problems

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

New analytical method for solving Burgers' and nonlinear heat transfer equations and comparison with HAM

In this study, we present a numerical comparison between the differential transform method (DTM) and the homotopy analysis method (HAM) for solving Burgers' and nonlinear heat transfer problems. The first differential equation is the Burgers' equation serves as a useful model for many interesting problems in applied mathematics. The second one is the modeling equation of a straight fin with a temperature dependent thermal conductivity. In order to show the effectiveness of the DTM, the results obtained from the DTM is compared with available solutions obtained using the HAM [M.M. Rashidi, G. Domairry, S. Dinarvand, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 708-717; G. Domairry, M. Fazeli, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 489-499] and whit exact solutions. The method can easily be applied to many linear and nonlinear problems. It illustrates the validity and the great potential of the differential transform method in solving nonlinear partial differential equations. The obtained results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations and nonlinear ordinary differential equations that we are found to be in good agreement with the exact solutions.     

An iterative multistep formula for solving initial value problems

The iterative multistep method (IMS) introduced by Hyman (1978) for solving initial value problems in ordinary differential equations has the advantage of being able to offer a higher degree of accuracy than the Runge-Kutta formulas by continuing the iteration process. In this article, another IMS formula is developed based on the geometric means predictor-corrector formulas introduced by Sanugi and Evans (1989). A numerical example is provided that shows that this formula can be used as a competitive alternative to Hyman's IMS formula.     

Sextic spline collocation methods for nonlinear fifth-order boundary value problems

In this paper, two sextic-spline collocation methods are developed and analysed for approximating solutions of nonlinear fifth-order boundary-value problems. The first method uses a spline interpolant and the second one is based on a spline quasi-interpolant, which are constructed from sextic splines. They are both proved to be second-order convergent. Numerical results confirm the order of convergence predicted by the analysis. It has been observed that the methods developed in this paper are better than the others given in the literature.