Bilal Chanane

Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method

The purpose in this paper is to compute the eigenvalues of Sturm-Liouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method, an improvement on the method based on Shannon sampling theory, which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost. A few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions.

Accurate solutions of fourth order Sturm-Liouville problems

Recently we introduced a new method which we call the Extended Sampling Method to compute the eigenvalues of second order Sturm-Liouville problems with eigenvalue dependent potential. We shall see in this paper how we use this method to compute the eigenvalues of fourth order Sturm-Liouville problems and present its practical use on a few examples.

Computing the spectrum of non-self-adjoint Sturm-Liouville problems with parameter-dependent boundary conditions

This paper deals with the computation of the eigenvalues of non-self-adjoint Sturm-Liouville problems with parameter-dependent boundary conditions using the regularized sampling method. A few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available.

Eigenvalues of Sturm-Liouville problems with discontinuity conditions inside a finite interval

Abstract In this work, we use the regularized sampling method to compute the eigenvalues of Sturm–Liouville problems with discontinuity conditions inside a finite interval. We work out an example by computing a few eigenvalues and their corresponding eigenfunctions.

Computing the eigenvalues of singular Sturm-Liouville problems using the regularized sampling method

Abstract This paper deals with singular Sturm–Liouville problems. We shall extend the domain of application of the regularized sampling method, a method to compute the eigenvalues. A few numerical examples will be presented to illustrate the merit of the method.

Effect of spatial side-wall temperature variation on transient natural convection of a nanofluid in a trapezoidal cavity

Purpose This paper aims to study analytically and numerically the problem of transient natural convection heat transfer in a trapezoidal cavity with spatial side-wall temperature variation. Design/methodology/approach The governing equations subject to the initial and boundary conditions are solved numerically by the finite difference scheme consisting of the alternating direction implicit method and the tri-diagonal matrix algorithm. The left sloping wall of the cavity is heated to non-uniform temperature, and the right sloping wall is maintained at a constant cold temperature, while the horizontal walls are kept adiabatic. Findings It is shown that the heat transfer rate increases in non-uniform heating increments, whereby low wave number values are more affected by the convection. The best heat transfer enhancement results from larger side wall inclination angle; however, trapezoidal cavities require longer time compared to that of square to reach steady state. Originality/value The study of natural convection heat transfer in a trapezoidal cavity filled with nanofluid and heated by spatial side-wall temperature has not yet been undertaken. Thus, the authors of the present study believe that this work is valuable.

Time-fractional nonlinear gas transport equation in tight porous media: An application in unconventional gas reservoirs

The prospects of meeting the futures high energy demands lie in the exploration of unconventional hydrocarbon reservoirs, of which the shale gas and the tight gas are two important resources. The deep understanding of such reservoirs is crucial to the economical recovery of such energy resources. With the advancement in the technological sides, such as, hydraulic fracturing and horizontal drilling, new mathematical models are needed that can precisely capture the complexity of the physical phenomena and can describe the flow of gas through the natural and induced fractures. The performance and future behavior of such reservoirs can be enhanced through careful modeling. We develop a new mathematical model based on time fractional derivative combined with the consideration of various flow regimes and a nonlinear treatment of reservoir parameters. The model describes the transport of gas in tight porous media (such as shale formations). The derivation of the model is done by using the mass balance equation and momentum conservation equation (basically modified time-fractional form of Darcys law) which incorporates the properties of tight porous media and accounts on the previous behavior. We find the pressure equation by considering that the rock properties, such as, permeability, viscosity, porosity, are pressure dependent. The pressure equation can be used to study the pressure distribution in the reservoir.

Computation of the Eigenpairs of Two-Parameter Sturm-Liouville Problems Using the Regularized Sampling Method

This paper deals with the computation of the eigenvalues of two-parameter Sturm-Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (singular, non-self-adjoint, nonlocal, impulsive, etc.). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.

Solutions of a class of partial differential equations with application to the Black–Scholes equation

Abstract In this paper we shall derive the solutions of a class of partial differential equations and its application to the Black–Scholes equation.

A new functional expansion for nonlinear systems

Abstract In this paper, we shall introduce a new functional series expansion of the output of a general nonlinear system as power series in powers of derivatives of the input components. We shall illustrate the usefulness of this series by considering two important problems namely those of “realization” and systems inversion.