فراس محمود سليمان الفقيه

Space time integration plays an important role in analyzing scientific and engineering models. In this paper, we consider an integrodifferential equation that comes from modeling neuron networks. Here, we investigate various schemes for time discretization of a theta-neuron model. We use collocation and midpoint quadrature formula for space integration and then apply various time integration schemes to get a full discrete system. We present some computational results to demonstrate the schemes.

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejér points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized Hölder spaces.

The main proposal of this article is the investigation and theoretical background of the direct-approximate methods for the numerical solution of singular integro-differential(SIDE) equations(Cauchy type kernel) with unknown function defined on the smooth contours of the Lypunov type. The equations are studied in the Lebesgue spaces. The SIDE are defined on the displaced Fej´er points of complex plane. The numerical schemes of collocation and mechanical quadrature methods for the SIDE defined on an arbitrary smooth closed contour of complex plane are elaborated. The theorems of convergence of these methods have been proved in Lebesgue spaces.

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fej�r points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized H�lder spaces.

An exact solution of the Three-Wave-Interaction (TW I) equations is well known by a compact formula called the N−soliton solution, a deep look to this solution shows that we need to find the inverse of some matrix whose entries are long formula of functions depend on time t and one spatial dimension x, which becomes very complicated for large N. An interested physical problem is to study the analytic form of the solution of the (TW I) for large N, in order to build what is called a train of soliton solutions, which will enable us to construct a pulse of a lot number of humps. Here, we simplified the form of the (N +1)− soliton solutions, and wrote it in terms of the N− soliton solutions plus some extra terms, then we approximated these terms and made them very simple. This will enable us to build the train of soliton solutions one by one by simply add these terms each time. We also examined successfully the (N +1)−soliton solutions for small values of N analytically and graphically

Computational schemes for the collocation method and the mechanical quadrature method for the approximate solution of systems of singular integro-differential equations with a Cauchy kernel are elaborated. The case where the systems of equations are defined on an arbitrary smooth closed contour of a complex plane is examined. The methods researched are based on Fejér points. Estimates of the rate of convergence in Lebesgue space are obtained.

The computation schemes of spline-collocation methods for solving singular integral equations. A theoretical foundation of these two methods is obtained in space L2. In the present paper we give theoreticaly justification of the numerical schemes of spline-collocation method for solving the singular integral equations (SIE) of the following form

We obtain the numerical schemes of collocation methods and mechanical quadratic methods to approximate the solutions of the singular integral equations. The equations are defined on the arbitrary smooth closed contour of the complex plane. Theoretical background for these methods is proved in classical H¨older spaces in the case when singular integral equations have finite number of different zeros in pairwise.

This study aims at considering the properties of heat transfer and magneto-hydrodynamics (MHD) Casson nanofluid at the existence of free convection boundary layer flow with Carbon Nanotubes (CNTs) suspended in human blood/water as based fluid on a stretching sheet. Two types of CNTs nanoparticles, single walled carbon nanotubes (SWCNTs) and multi walled carbon nanotubes (MWCNTs), are taken into account. The governing partial differential equations are transformed to partial differential equations using similar transformation, then solved numerically by an implicit finite difference scheme known as Keller-box method (KBM). The results for physical quantities, the local skin friction, and local Nusselt number, as well as temperature and velocity, are discussed under the magnetic nanofluid Casson parameters. This work is compared with recently published results on the Newtonian fluid as a special case and shows very good agreement.

The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

During the last decades, no researches have conducted in order to prove some properties of the of the multivariate power series distribution, as results of the present study proved that any multivariate power series distribution is determined uniquely from the mean –function of any marginal random variable. Furthermore these results indicated also that any given function satisfying certain conditions construct a random vector with multivariate power series distribution which has a mean of the marginal random variable. A useful technique can be applied in model building when we have information about the mean- function.

For a simple graph G, a map φ : V (G) → {1, 2, . . . , k} is called a vertex k-labeling. For any edge vu in G, its weight φ(vu) = φ(v) + φ(u). If all of the edges weights are distinct, then φ is called an edge irregular k-labeling of G. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine an exact value of edge irregularity strength of complete bipartite graph Kn,2, corona product of Pn with P6 and Pn with C3.

The paper must have abstract not exceeding 200 words. In this paper, we shall investigate some mathematical difficulties that is facing learners and undergraduates in science and engineering when they use MATLAB. When solving some mathematical problems using this software, one may unfortunately, face the following: no answer at all, wrong answer, long and complicated answer. Therefore, our main aim is to investigate such problems through mathematical examples in order for the user to be aware of.

: In this article we have elaborated the numerical schemes of reduction methods for approximate solution of system of singular integro-differential equations when the kernel has a weak singularity. The equations are defined on the arbitrary smooth closed contour of complex plane. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for the approximate solution of weakly singular integro- differential equations defined on smooth closed contours in the complex plane. We use the cut-off technique kernel to reduce the weakly singular integro- differential equation to the continuous one. Our approach is based on the Krykunov theory and Zolotarevski results. We have obtained the theoretical background for these methods in classical Lebesgue spaces.

Space time integration plays an important role in analyzing scientific and engineering models. In this paper, we consider an integrodifferential equation that comes from modeling neuron networks. Here, we investigate various schemes for time discretization of a theta-neuron model. We use collocation and midpoint quadrature formula for space integration and then apply various time integration schemes to get a full discrete system. We present some computational results to demonstrate the schemes.

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fejér points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized Hölder spaces.

The main proposal of this article is the investigation and theoretical background of the direct-approximate methods for the numerical solution of singular integro-differential(SIDE) equations(Cauchy type kernel) with unknown function defined on the smooth contours of the Lypunov type. The equations are studied in the Lebesgue spaces. The SIDE are defined on the displaced Fej´er points of complex plane. The numerical schemes of collocation and mechanical quadrature methods for the SIDE defined on an arbitrary smooth closed contour of complex plane are elaborated. The theorems of convergence of these methods have been proved in Lebesgue spaces.

We have elaborated the numerical schemes of collocation methods and mechanical quadrature methods for approximate solution of singular integro- differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on Fej�r points. Theoretical background of collocation methods and mechanical quadrature methods has been obtained in Generalized H�lder spaces.

An exact solution of the Three-Wave-Interaction (TW I) equations is well known by a compact formula called the N−soliton solution, a deep look to this solution shows that we need to find the inverse of some matrix whose entries are long formula of functions depend on time t and one spatial dimension x, which becomes very complicated for large N. An interested physical problem is to study the analytic form of the solution of the (TW I) for large N, in order to build what is called a train of soliton solutions, which will enable us to construct a pulse of a lot number of humps. Here, we simplified the form of the (N +1)− soliton solutions, and wrote it in terms of the N− soliton solutions plus some extra terms, then we approximated these terms and made them very simple. This will enable us to build the train of soliton solutions one by one by simply add these terms each time. We also examined successfully the (N +1)−soliton solutions for small values of N analytically and graphically

Computational schemes for the collocation method and the mechanical quadrature method for the approximate solution of systems of singular integro-differential equations with a Cauchy kernel are elaborated. The case where the systems of equations are defined on an arbitrary smooth closed contour of a complex plane is examined. The methods researched are based on Fejér points. Estimates of the rate of convergence in Lebesgue space are obtained.

The computation schemes of spline-collocation methods for solving singular integral equations. A theoretical foundation of these two methods is obtained in space L2. In the present paper we give theoreticaly justification of the numerical schemes of spline-collocation method for solving the singular integral equations (SIE) of the following form

We obtain the numerical schemes of collocation methods and mechanical quadratic methods to approximate the solutions of the singular integral equations. The equations are defined on the arbitrary smooth closed contour of the complex plane. Theoretical background for these methods is proved in classical H¨older spaces in the case when singular integral equations have finite number of different zeros in pairwise.

This study aims at considering the properties of heat transfer and magneto-hydrodynamics (MHD) Casson nanofluid at the existence of free convection boundary layer flow with Carbon Nanotubes (CNTs) suspended in human blood/water as based fluid on a stretching sheet. Two types of CNTs nanoparticles, single walled carbon nanotubes (SWCNTs) and multi walled carbon nanotubes (MWCNTs), are taken into account. The governing partial differential equations are transformed to partial differential equations using similar transformation, then solved numerically by an implicit finite difference scheme known as Keller-box method (KBM). The results for physical quantities, the local skin friction, and local Nusselt number, as well as temperature and velocity, are discussed under the magnetic nanofluid Casson parameters. This work is compared with recently published results on the Newtonian fluid as a special case and shows very good agreement.

The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

During the last decades, no researches have conducted in order to prove some properties of the of the multivariate power series distribution, as results of the present study proved that any multivariate power series distribution is determined uniquely from the mean –function of any marginal random variable. Furthermore these results indicated also that any given function satisfying certain conditions construct a random vector with multivariate power series distribution which has a mean of the marginal random variable. A useful technique can be applied in model building when we have information about the mean- function.

For a simple graph G, a map φ : V (G) → {1, 2, . . . , k} is called a vertex k-labeling. For any edge vu in G, its weight φ(vu) = φ(v) + φ(u). If all of the edges weights are distinct, then φ is called an edge irregular k-labeling of G. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine an exact value of edge irregularity strength of complete bipartite graph Kn,2, corona product of Pn with P6 and Pn with C3.

The paper must have abstract not exceeding 200 words. In this paper, we shall investigate some mathematical difficulties that is facing learners and undergraduates in science and engineering when they use MATLAB. When solving some mathematical problems using this software, one may unfortunately, face the following: no answer at all, wrong answer, long and complicated answer. Therefore, our main aim is to investigate such problems through mathematical examples in order for the user to be aware of.

: In this article we have elaborated the numerical schemes of reduction methods for approximate solution of system of singular integro-differential equations when the kernel has a weak singularity. The equations are defined on the arbitrary smooth closed contour of complex plane. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for the approximate solution of weakly singular integro- differential equations defined on smooth closed contours in the complex plane. We use the cut-off technique kernel to reduce the weakly singular integro- differential equation to the continuous one. Our approach is based on the Krykunov theory and Zolotarevski results. We have obtained the theoretical background for these methods in classical Lebesgue spaces.

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                SAS , Minitab & MATLAB