Mohamed Ali Hajji

An efficient algorithm for solving higher-order fractional Sturm–Liouville eigenvalue problems

Abstract In this paper, we present a simple and efficient computational algorithm for solving eigenvalue problems of high fractional-order differential equations with variable coefficients. The method of solution is based on utilizing the series solution to convert the governing fractional differential equation into a linear system of algebraic equations. Then, the eigenvalues can be calculated by finding the roots of the corresponding characteristic polynomial. Notice that this class of eigenvalue problems is very promising to the solution of linear fractional partial differential equations (FPDE). The numerical results demonstrate reliability and efficiency of the proposed algorithm. Based on our simulations some theoretical conjectures are reported.

Analytic studies and numerical simulations of the generalized Boussinesq equation

The modified Adomian decomposition method is used to solve the generalized Boussinesq equation. The equation commonly describes the propagation of small amplitude long waves in several physical contents. The analytic solution of the equation is obtained in the form of a convergent series with easily computable components. For comparison purposes, a numerical algorithm, based on Chebyshev polynomials, is developed and simulated. Numerical results show that the modified Adomian decomposition method proves to be more accurate and computationally more efficient than the Galerkin-Chebyshev method.

A Convergent Algorithm for Solving Higher-Order Nonlinear Fractional Boundary Value Problems

Abstract We present a numerical algorithm for solving nonlinear fractional boundary value problems of order n, n ∈ ℕ. The Bernstein polynomials (BPs) are redefined in a fractional form over an arbitrary interval [a, b]. Theoretical results related to the ractional Bernstein polynomials (FBPs) are proven. The well-known shooting technique is extended for the numerical treatment of nonlinear fractional boundary value problems of arbitrary order. The initial value problems were solved using a collocation method with collocation points at the location of the local maximum of the FBPs. Several examples are discussed to illustrate the efficiency and accuracy of the present scheme.

Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo ( ) and Riemann ( ) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order is proved and the ordinary difference Lyapunov inequality then follows as tends to from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

An Efficient Series Solution for Fractional Differential Equations

We introduce a simple and efficient series solution for a class of nonlinear fractional differential equations of Caputos type. The new approach is a modified form of the well-known Taylor series expansion where we overcome the difficulty of computing iterated fractional derivatives, which do not compute in general. The terms of the series are determined sequentially with explicit formula, where only integer derivatives have to be computed. The efficiency of the new algorithm is illustrated through several examples. Comparison with other series methods such as the Adomian decomposition method and the homotopy perturbation method is made to indicate the efficiency of the new approach. The algorithm can be implemented for a wide class of fractional differential equations with different types of fractional derivatives.

On the similarity solution of nano-fluid flow over a moving flat plate using the homotopy analysis method

The present article discusses the characteristics of Newtonian water-base-Copper nano-fluid flowing over an infinite flat plate moving with a constant velocity in the direction of the flow. The non-classical similarity transformation is employed to transform the Navier-Stokes equation into a nonlinear ordinary differential equation with specific boundary conditions. The Homptopy analysis method (HAM) is employed to solve the resulting nonlinear differential equation to study the effects of the nanoparticle volume fraction and wall velocity on the flow velocity profile, the boundary layer thickness and the local skin friction coefficient. The existence and non-uniqueness of the solution as a function of the wall velocity will be also discussed.

A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

Abstract In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton’s method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.

Numerical methods for nonlinear fourth-order boundary value problems with applications

In this paper, we present efficient numerical algorithms for the approximate solution of nonlinear fourth-order boundary value problems. The first algorithm deals with the sinc–Galerkin method (SGM). The sinc basis functions prove to handle well singularities in the problem. The resulting SGM discrete system is carefully developed. The second method, the Adomian decomposition method (ADM), gives the solution in the form of a series solution. A modified form of the ADM based on the use of the Laplace transform is also presented. We refer to this method as the Laplace Adomian decomposition technique (LADT). The proposed LADT can make the Adomian series solution convergent in the Laplace domain, when the ADM series solution diverges in the space domain. A number of examples are considered to investigate the reliability and efficiency of each method. Numerical results show that the sinc–Galerkin method is more reliable and more accurate.

Solving nonlinear boundary value problems using the homotopy analysis method

Homotopy analysis method (HAM) has been employed recently by many authors to solve nonlinear problems, in particular nonlinear initial and boundary values problems. Such nonlinear problems are usually derived from physical problems such as fluid mechanics; heat transfer, boundary layer equations and many others. In the suggested work we will extend the use of the HAM to solve a certain class of boundary value problems. Focus will be on multi-layer boundary problems. Examples of these kind of problems include fluid flow through multi-layer porous media.

Two reliable methods for solving nonlinear evolution equations

This paper aims to introduce a comparison of Adomian decomposition and wavelet basis method for the numerical solution of nonlinear evolution equations. Test problems concerning the motion and interaction of solutions are used to compare between the two methods. The comparisons shows that the Adomian decomposition method, in terms of accuracy, is efficient and easy to use.