Qasem M. Al-Mdallal

On the numerical solution of fractional Sturm-Liouville problems

The differential equation of Sturm-Liouville problems is generalized into fractional form by replacing the first-order derivative by a fractional derivative of order α, 0<α≤1. We showed briefly that this class of eigenvalue could be very promising to the solution of linear fractional partial differential equations. The homotopy perturbation method is considered for computing the eigenelements of the present problem. Based on our simulations some theoretical conjectures are reported.

On the asymptotic stability of linear system of fractional-order difference equations

In this paper we investigate the stability of the equilibrium solution of the νth order linear system of difference equations

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

(\Delta _{a + \nu - 1}^\nu y)(t) = \Lambda y(t + \nu - 1);t \in \mathbb{N}_a ,a \in \mathbb{R},and\Lambda \in \mathbb{R}^{p \times p} ,

Heat transfer enhancement in free convection flow of CNTs Maxwell nanofluids with four different types of molecular liquids

subject to the initial condition

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

y(a + \nu - 1) = y - 1,

The Chebyshev collocation-path following method for solving sixth-order Sturm-Liouville problems

, where 0 < ν < 1 and y−1 ∈ ℝp.

Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

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.

The extended homotopy perturbation method for the boundary layer flow due to a stretching sheet with partial slip

This article investigates heat transfer enhancement in free convection flow of Maxwell nanofluids with carbon nanotubes (CNTs) over a vertically static plate with constant wall temperature. Two kinds of CNTs i.e. single walls carbon nanotubes (SWCNTs) and multiple walls carbon nanotubes (MWCNTs) are suspended in four different types of base liquids (Kerosene oil, Engine oil, water and ethylene glycol). Kerosene oil-based nanofluids are given a special consideration due to their higher thermal conductivities, unique properties and applications. The problem is modelled in terms of PDE’s with initial and boundary conditions. Some relevant non-dimensional variables are inserted in order to transmute the governing problem into dimensionless form. The resulting problem is solved via Laplace transform technique and exact solutions for velocity, shear stress and temperature are acquired. These solutions are significantly controlled by the variations of parameters including the relaxation time, Prandtl number, Grashof number and nanoparticles volume fraction. Velocity and temperature increases with elevation in Grashof number while Shear stress minimizes with increasing Maxwell parameter. A comparison between SWCNTs and MWCNTs in each case is made. Moreover, a graph showing the comparison amongst four different types of nanofluids for both CNTs is also plotted.

The extended homotopy perturbation method and boundary layer flow due to condensation and natural convection on a porous vertical plate

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.

Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality

In this paper, we implement a Chebyshev collocation method to approximate the eigenvalues of nonsingular sixth-order Sturm–Liouville problem. This method transforms the Sturm–Liouville problem to a sparse singular linear system which is solved by the path following technique. Numerical results demonstrate the accuracy and efficiency of the present algorithm.