Basem S. Attili’s research while affiliated with University of Sharjah and other places

We consider a singularly perturbed one-dimensional reaction-diffusion three-point boundary value problem. To approximate the solution numerically, we employ an exponentially fitted finite uniform difference scheme defined on a piecewise uniform Shishkin mesh which is second order and uniformly convergent independent of the perturbation parameter. We will present some numerical examples to show the efficiency of the proposed method.

We will consider the numerical solution of the Hessenberg linear index-2 differential algebraic equations (DAE’s) using the differential transform method. We will first present a method to reduce the index of the DAE to index-1. This results in a problem easier to solve. Numerical examples in addition to the comparison with the work of others will also be presented to show the efficiency of the method.

We will numerically investigate the wave instability problem with the effect of the transverse velocity component. An accurate and easy to program finite difference scheme will be developed for this purpose. The eigenfunctions will be normalized and computed simultaneously with the eigenvectors. Numerical results will also be presented.

A boundary value method for solving a class of nonlinear singularly perturbed two point boundary value problems with a boundary layer at one end is proposed. Using singular perturbation analysis the method consists of solving two problems; namely, a reduced problem and a boundary layer correction problem. We use Pade’ approximation to obtain the solution of the latter problem and to satisfy the condition at infinity. Numerical examples will be given to illustrate the method.

We will consider the Hilber–Hughes–Taylor-α (HHT-α ) method to solve periodic second-order initial value problems arising in, e.g. mechanics. We will consider the analysis of the method when applied to such problems. Second-order convergence is theoretically demonstrated and numerically illustrated. In addition, we will consider the efficient implementation of an implicit fourth-order Runge–Kutta scheme. Numerical details and examples will also be presented to demonstrate the efficiency of the methods.

A numerical algorithm based on a second order finite difference scheme is proposed to solve singularly perturbed linear two point boundary value problems. The system is first decoupled partially into two independent subsystems, fast and slow components. Each subsystem then is solved separately using the proposed algorithm. We present some numerical examples to show the efficiency of the algorithm.

A direct method for the numerical computation of pitchfork bifurcation points under certain symmetry conditions is presented. We will be interested in computing the critical parameter. The direct method presented produces a larger system of full rank and hence solvable. The presence of symmetry will be of help since it will reduce the amount of work needed. Numerical experimentation will be done to demonstrate the efficiency of the suggested approach. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

We will consider the numerical computation of pitchfork bifurcation points under certain symmetry conditions for parameter dependent problems. These types of bifurcation points are referred to as symmetry breaking bifurcation points. A direct method is employed to characterize the singularity. The presence of symmetry will reduce the amount of work needed. Numerical examples will be solved to demonstrate the efficiency of the suggested approach.

We present an efficient shooting method for solving two point boundary value problems. The Adomian decomposition method will be utilized to obtain a series solution of the initial value problems involved. Numerical examples and comparison of the work of others will also be done.

Following earlier works on second- and fourth-order problems, we develop an efficient method based on the Adomian decomposition for computing the eigenelements of sixth-order Sturm-Liouville boundary value problems. Numerical examples show that the method proposed is easy to implement and produces accurate results.