Mohammed A. Alghamdi

A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals

In this paper, we develop a Jacobi-Gauss-Lobatto collocation method for solving the nonlinear fractional Langevin equation with three-point boundary conditions. The fractional derivative is described in the Caputo sense. The shifted Jacobi-Gauss-Lobatto points are used as collocation nodes. The main characteristic behind the Jacobi-Gauss-Lobatto collocation approach is that it reduces such a problem to those of solving a system of algebraic equations. This system is written in a compact matrix form. Through several numerical examples, we evaluate the accuracy and performance of the proposed method. The method is easy to implement and yields very accurate results.

On the interplay between mathematics and biology: hallmarks toward a new systems biology.

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.

Fixed point and coupled fixed point theorems on b-metric-like spaces

We first introduce the concept of b-metric-like space which generalizes the notions of partial metric space, metric-like space and b-metric space. Then we establish the existence and uniqueness of fixed points in a b-metric-like space as well as in a partially ordered b-metric-like space. As an application, we derive some new fixed point and coupled fixed point results in partial metric spaces, metric-like spaces and b-metric spaces. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the obtained results.MSC:47H10, 54H25, 55M20.

Statistical summability through a lacunary sequence in locally solid Riesz spaces

In this paper, we introduce the concepts of lacunary statistical τ-convergence, lacunary statistically τ-bounded and lacunary statistically τ-Cauchy in the framework of locally solid Riesz spaces. We also define a new type of convergence, that is, S∗(τ)-convergence in this setup and prove some interesting results related to these notions.MSC:40A35, 40G15, 46A40.

A Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays

In this paper, we present a unified framework for analyzing the spectral collocation method for neutral functional-differential equations with proportional delays using shifted Legendre polynomials. The proposed collocation technique is based on shifted Legendre-Gauss quadrature nodes as collocation knots. Error analysis and stability of the proposed algorithm are theoretically investigated under several mild conditions. The accuracy of the proposed method has been compared with a variational iteration method, a one-leg θ-method, a particular Runge-Kutta method, and a reproducing kernel Hilbert space method. Numerical results show that the proposed methods are of high accuracy and are efficient for solving such an equation. Also, the results demonstrate that the proposed method is a powerful algorithm for solving other delay differential equations.

Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces

In this paper, we determine the stability of a generalized Hyers-Ulam-Rassias-type theorem concerning the additive functional equation 2f(x+y+z2)=f(x)+f(y)+f(z) in the framework of intuitionistic fuzzy normed spaces through the fixed-point alternative. Further, we prove some stability results of an additive functional equation in this setup through the direct method.MSC:39B52, 39B82, 46S40.

Best proximity point results in geodesic metric spaces

In this paper, the existence of a best proximity point for relatively u-continuous mappings is proved in geodesic metric spaces. As an application, we discuss the existence of common best proximity points for a family of not necessarily commuting relatively u-continuous mappings.

The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications

In this paper, the modified generalized Laguerre operational matrix (MGLOM) of Caputo fractional derivatives is constructed and implemented in combination with the spectral tau method for solving linear multi-term FDEs on the half-line. In this approach, truncated modified generalized Laguerre polynomials (MGLP) together with the modified generalized Laguerre operational matrix of Caputo fractional derivatives are analyzed and applied for numerical integration of such equations subject to initial conditions. The modified generalized Laguerre pseudo-spectral approximation based on the modified generalized Laguerre operational matrix is investigated to reduce the nonlinear multi-term FDEs and their initial conditions to a nonlinear algebraic system. Through some numerical experiments, we evaluate the accuracy and efficiency of the proposed methods. The methods are easy to implement and yield very accurate results.

New upper bounds of n

AbstractIn this article, we deduce a new family of upper bounds of n! of the form n!<2πn(n/e)neMn[m]n∈ℕ,Mn[m]=12m+314n+∑k=1m2m-2k+22k+12-2kζ(2k,n+1/2)m=1,2,3,.... We also proved that the approximation formula 2πn(n/e)neMn[m] for big factorials has a speed of convergence equal to n-2m- 3for m = 1,2,3,..., which give us a superiority over other known formulas by a suitable choice of m.Mathematics Subject Classification (2000): 41A60; 41A25; 57Q55; 33B15; 26D07.

Hankel matrix transformation of the Walsh-Fourier series

The Hankel matrix has various applications. In this paper we prove that Hankel matrix is strongly regular and apply to obtain the necessary and sufficient conditions to sum the Walsh-Fourier series of a function of bounded variation.