An optimization method for studying fractional-order tuberculosis disease model via generalized Laguerre polynomials.

Tuberculosis (TB) is a deadly contagious disease that affects vital organs of the body, especially the lungs. Although the disease is preventable, there are still concerns about its continued spread. Without effective prevention or appropriate treatment, TB infection can be fatal to humans. This paper presents a fractional-order TB disease (FTBD) model to analyze TB dynamics and a new optimization method to solve it. The method is based on the basis functions of generalized Laguerre polynomials (GLPs) and some new operational matrices of derivatives in the Caputo sense. Finding the optimal solution to the FTBD model is reduced to solving a system of nonlinear algebraic equations with the aid of GLPs using the Lagrange multipliers method. A numerical simulation is also carried out to determine the impact of the presented method on the susceptible, exposed, infected without treatment, infected with treatment, and recovered cases in the population.

A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer.

INTRODUCTION: During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. OBJECTIVES: The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann-Liouville sense. METHODS: The temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. RESULTS: Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique. CONCLUSION: This paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.

Optimal solution of the fractional order breast cancer competition model.

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics.

This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative. The model is often adopted for describing the anomalous process of nearly sonic speed gas dynamics. The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode. The local collocation method approximates the differential operators using a weighted sum of the function values over a local collection of nodes (named stencil) through a radial basis function expansion. This technique considers merely the discretization nodes of each subdomain around the collocation node. This leads to sparse systems and tackles the ill-conditioning produced of global collocation. The theoretical convergence and stability analyses of the proposed time semi-discrete scheme are proved by means of the discrete energy method. Numerical results confirm the accuracy and efficiency of the new approach.

Gravitational Field effects on the Decoherence Process and the Quantum Speed Limit.

In this paper we use spinor transformations under local Lorentz transformations to investigate the curvature effect on the quantum-to-classical transition, described in terms of the decoherence process and of the quantum speed limit. We find that gravitational fields (introduced adopting the Schwarzschild and anti-de Sitter geometries) affect both the decoherence process and the quantum speed limit of a quantum particle with spin-1/2. In addition, as a tangible example, we study the effect of the Earth's gravitational field, characterized by the Rindler space-time, on the same particle. We find that the effect of the Earth's gravitational field on the decoherence process and quantum speed limit is very small, except when the mean speed of the quantum particle is comparable to the speed of light.