Browsing by Author "Ali, Mubasher"

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Numerical analysis of diffusive susceptible-infected-recovered epidemic model in three space dimension
(2020) Baleanu, Dumitru; Ali, Mubasher; Baleanu, Dumitru; Rafiq, Muhammad; Rehman, Muhammad Aziz Ur; 56389
In this article, numerical solution of three dimensional susceptible-infected-recovered (SIR) reaction-diffusion epidemic system is furnished with a time efficient operator splitting nonstandard finite difference (OS-NSFD) method. We perform the comparison of proposed OS-NSFD method with popular forward Euler explicit finite difference (FD) method and time efficient backward Euler operator splitting finite difference (OS-FD) implicit method. The proposed OS-NSFD method is implicit in nature but computationally efficient as compared to forward Euler explicit (FD) scheme. The numerical stability and bifurcation value of transmission coefficient for SIR reaction-diffusion epidemic system is also investigated with the aid of Routh-Hurwitz method. At the end, we give two numerical experiments and simulation. In first experiment, all the numerical schemes are compared with the help of simulations. In second experiment we show the simulations of proposed NSFD technique at different values of parameters. Also we discuss the importance of transmission rate to control the spread of disease with the help of simulations. (C) 2019 Elsevier Ltd. All rights reserved.
Positivity preserving computational techniques for nonlinear autocatalytic chemical reaction model
(2020) Baleanu, Dumitru; Baleanu, Dumitru; Korkmaz, Alper; Rafiq, Muhammad; Rehman, Muhammad Aziz-Ur; Ali, Mubasher; 56389
In many physical problems, positivity is one of the most prevalent and imperative attribute of diverse mathematical models such as concentration of chemical reactions, population dynamics etc. However, the numerical discretization of dynamical systems that illustrate negative values may lead to meaningless solutions and sometimes to their divergence. The main objective of this work is to develop positivity preserving numerical schemes for the two-dimensional autocatalytic reaction diffusion Brusselator model. Two explicit finite difference (FD) schemes are proposed to solve numerically the two-dimensional Brusselator system. The proposed methods are the non-standard finite difference (NSFD) scheme and the unconditionally positivity preserving scheme. These numerical methods retain the positivity of the solution and the stability of the equilibrium point. Both proposed numerical schemes are compared with the forward Euler explicit FD scheme. The stability and consistency of all schemes are proved analytically and then verified by numerical simulations. © 2020, Editura Academiei Romane. All rights reserved.
Structure preserving algorithms for mathematical model of auto-catalytic glycolysis chemical reaction and numerical simulations
(2020) Baleanu, Dumitru; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-u; Khan, Ilyas; Ali, Mubasher; Nisar, Kottakkaran Sooppy; 56389
This paper aims to develop positivity preserving splitting techniques for glycolysis reaction-diffusion chemical model. The positivity of state variables in the glycolysis model is an essential property that must be preserved for all choices of parameters. We propose two splitting methods that remain dynamically consistent with the continuous glycolysis reaction-diffusion model. The proposed methods converge to a true steady-state or fixed point under the given condition. On contrary to the classical operator splitting finite difference methods, we use nonstandard finite difference theory to propose a new class of operator splitting techniques.
Structure Preserving Numerical Analysis of Reaction-Diffusion Models
(2022) Jarad, Fahd; Muhammad Aziz-Ur, Rehman; Waleed, Adel; Jarad, Fahd; Ali, Mubasher; Rafiq, Muhammad; Akgül, Ali; 234808
In this paper, we examine two structure preserving numerical finite difference methods for solving the various reaction-diffusion models in one dimension, appearing in chemistry and biology. These are the finite difference methods in splitting environment, namely, operator splitting nonstandard finite difference (OS-NSFD) methods that effectively deal with nonlinearity in the models and computationally efficient. Positivity of both the proposed splitting methods is proved mathematically and verified with the simulations. A comparison is made between proposed OS-NSFD methods and well-known classical operator splitting finite difference (OS-FD) methods, which demonstrates the advantages of proposed methods. Furthermore, we applied proposed NSFD splitting methods on several numerical examples to validate all the attributes of the proposed numerical designs.