Ali Sayfy

A novel approach for the solution of a class of singular boundary value problems arising in physiology

A new approach implementing a modified decomposition method in combination with the cubic B-spline collocation technique is introduced for the numerical solution of a class of singular boundary value problems arising in physiology. The domain of the problem is split into two subintervals; a modified decomposition procedure based on a special integral operator is implemented in the vicinity of the singular point and outside this domain the resulting boundary value problem is tackled by applying the B-spline scheme. Performance of this method is examined numerically; the examples reveal that the current approach converges to the exact solution rapidly and with O(h^2) accuracy. Results show that the method yields a numerical solution in very good agreement with the existing exact and approximate solutions.

Troesch’s problem: A B-spline collocation approach

Abstract A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solved using an adaptive spline collocation approach over a non-uniform mesh via exploiting an iterative scheme arising from Newton’s method. The convergence analysis is discussed and is shown to depend on the eigenvalues; in particular, the rate of convergence is calculated using the double-mesh principle. To demonstrate the efficiency of the method, a number of special cases are considered. The numerical solutions are compared with both the analytical solutions and other existing numerical solutions in the literature. It is observed that the results obtained by this method are quite satisfactory and accurate, and the method is applicable for a wide range of cases when contrasted with other available solutions.

A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation

In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein-Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels.

A Laplace variational iteration strategy for the solution of differential equations

Abstract The aim of this article is to introduce a novel Laplace variational numerical scheme, based on the variational iteration method (VIM) and Laplace transform, for the solution of certain classes of linear and nonlinear differential equations. The strategy is outlined and then illustrated through a number of test examples. The results assert that this alternative approach yields accurate results, converges rapidly and handles impulse functions and the ones with discontinuities.

A numerical approach for solving an extended Fisher-Kolomogrov-Petrovskii-Piskunov equation

In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed.

A novel fixed point iteration method for the solution of third order boundary value problems

In this work, a new alternative uniformly convergent iterative scheme is presented and applied for the solution of an extended class of linear and nonlinear third order boundary value problems that arise in physical applications. The method is based on embedding Greens functions into well-known fixed point iterations, including Picards and Krasnoselskii-Manns schemes. Convergence of the numerical method is proved by manipulating the contraction principle. The effectiveness of the proposed approach is established by implementing it on several numerical examples, including linear and nonlinear third order boundary value problems. The results show highly accurate approximations when compared to exact and existing numerical solutions.

Variational iteration method: Green’s functions and fixed point iterations perspective

Abstract The aim of this article is to demonstrate that the variational iteration method “VIM” is in many instances a version of fixed point iteration methods such as Picard’s scheme. In a wide range of problems, the correction functional resulting from the VIM can be interpreted and/or formulated from well-known fixed point strategies using Green’s functions. A number of examples are included to assert the validity of our claim. The test problems include first and higher order initial value problems.

A novel fixed point scheme: Proper setting of variational iteration method for BVPs

Abstract The aim of this article is to introduce a novel approach based on embedding Green’s function into a tailored linear integral operator then apply a well-known fixed point iterative scheme, such as Picard’s and Mann’s, for the numerical solution of initial and boundary value problems (IVPs and BVPs). The strategy provides insight into the proper setting of the variational iteration method (VIM) for BVPs. A convergence analysis of the method is included. A number of examples consisting of second and third order BVPs are presented to demonstrate the efficiency and applicability of the scheme.

The boundary layer problem: A fourth-order adaptive collocation approach

A finite element approach, based on the cubic B-spline collocation, is presented for the numerical solution of a class of singularly perturbed two-point boundary value problems that possesses a boundary layer at one or two end points. Due to the existence of a layer, the problem is handled using an adaptive spline collocation approach constructed over a non-uniform Shishkin-like mesh, defined via a carefully selected generating function. To tackle the case of nonlinearity, if it exists, an iterative scheme arising from Newtons method is employed. The rate of convergence is verified to be of fourth-order and is calculated using the double-mesh principle. The efficiency and applicability of the method are demonstrated by applying it to a number of linear and nonlinear examples. The numerical solutions are compared with both analytical and other existing numerical solutions in the literature. The numerical results confirm that this method is superior when contrasted with other accessible approaches and yields more accurate solutions.

A spline collocation approach for a generalized wave equation subject to non-local conservation condition

Abstract In the present paper, a collocation finite element approach based on cubic splines is presented for the numerical solution of a generalized wave equation subject to non-local conservation condition. The efficiency, accuracy and stability of the method are assessed by applying it to a number of test problems. The results are compared with the existing closed-form solutions; the scheme demonstrates that the numerical outcomes are reliable and quite accurate when contrasted with the analytical solutions and an existing numerical method.