Browsing by Author "Rehman, Muhammad Aziz-Ur"

Now showing 1 - 4 of 4

Citation Count: Jawaz, Muhammad;...et.al. (2023). "Analysis and numerical effects of time-delayed rabies epidemic model with diffusion", International Journal of Nonlinear Sciences and Numerical Simulation, Vol.24, No.6, pp.2179-2194.
Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
(2023) Jawaz, Muhammad; Rehman, Muhammad Aziz-Ur; Ahmed, Nauman; Baleanu, Dumitru; Iqbal, Muhammad Sajid; Rafiq, Muhammad; Raza, Ali; 56389
The current work is devoted to investigating the disease dynamics and numerical modeling for the delay diffusion infectious rabies model. To this end, a non-linear diffusive rabies model with delay count is considered. Parameters involved in the model are also described. Equilibrium points of the model are determined and their role in studying the disease dynamics is identified. The basic reproduction number is also studied. Before going towards the numerical technique, the definite existence of the solution is ensured with the help of the Schauder fixed point theorem. A standard result for the uniqueness of the solution is also established. Mapping properties and relative compactness of the operator are studied. The proposed finite difference method is introduced by applying the rules defined by R.E. Mickens. Stability analysis of the proposed method is done by implementing the Von-Neumann method. Taylor's expansion approach is enforced to examine the consistency of the said method. All the important facts of the proposed numerical device are investigated by presenting the appropriate numerical test example and computer simulations. The effect of τ on infected individuals is also examined, graphically. Moreover, a fruitful conclusion of the study is submitted.
Citation Count: Fatima, Umbreen...et al. (2021). "Numerical study of computer virus reaction diffusion epidemic model", Computers, Materials and Continua, Vol. 66, No. 3, pp. 3183-3194.
Numerical study of computer virus reaction diffusion epidemic model
(2021) Fatima, Umbreen; Baleanu, Dumitru; Ahmed, Nauman; Azam, Shumaila; Raza, Ali; Rafiq, Muhammad; Rehman, Muhammad Aziz-Ur; 56389
Reaction–diffusion systems are mathematical models which link to several physical phenomena. The most common is the change in space and time of the meditation of one or more materials. Reaction–diffusion modeling is a substantial role in the modeling of computer propagation like infectious diseases. We investigated the transmission dynamics of the computer virus in which connected to each other through network globally. The current study devoted to the structure-preserving analysis of the computer propagation model. This manuscript is devoted to finding the numerical investigation of the reaction–diffusion computer virus epidemic model with the help of a reliable technique. The designed technique is finite difference scheme which sustains the important physical behavior of continuous model like the positivity of the dependent variables, the stability of the equilibria. The theoretical analysis of the proposed method like the positivity of the approximation, stability, and consistency is discussed in detail. A numerical example of simulations yields the authentication of the theoretical results of the designed technique.
Citation Count: Ahmed, Nauman...et al. (2020). "Positivity preserving computational techniques for nonlinear autocatalytic chemical reaction model", Romanian Reports in Physics, Vol. 72, No. 4, pp. 1-15.
Positivity preserving computational techniques for nonlinear autocatalytic chemical reaction model
(2020) Ahmed, Nauman; 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.
Citation Count: Ahmed, Nauman...et al. (2019). "Spatio-Temporal Numerical Modeling of Auto-Catalytic Brusselator Model", Romanian Journal of Physics, Vol. 64.
Spatio-Temporal Numerical Modeling of Auto-Catalytic Brusselator Model
(Editura Academiei Romane, 2019) Ahmed, Nauman; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-Ur; 56389
The main objective of this article is to propose a chaos free explicit finite-difference (FD) scheme to find the numerical solution for the Brusselator reaction-diffusion model. The scheme is unconditionally stable and it is unconditionally dynamically consistent with the positivity property of continuous model as unknown quantities of auto-catalytic Brusselator system describe the concentrations of two reactant substances. Stability of the proposed FD method is showed with the help of Neumann criteria of stability. Taylor series is used to validate the consistency of the proposed FD method. Forward Euler explicit FD approach and semi-implicit Crank-Nicolson FD scheme are also applied to solve the Brusselator reaction-diffusion system and to make the comparison with the proposed FD scheme.