Browsing by Author "Rehman, Muhammad Aziz-ur"

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Citation Count: Iqbal, Zafar...et al. (2023). "A finite difference scheme to solve a fractional order epidemic model of computer virus", Aims Mathematics, Vol.8, No. 1, pp.2337-2359.
A finite difference scheme to solve a fractional order epidemic model of computer virus
(2023) Iqbal, Zafar; Rehman, Muhammad Aziz-ur; Imran, Muhammad; Ahmed, Nauman; Fatima, Umbreen; Akgul, Ali; Rafiq, Muhammad; Raza, Ali; Djuraev, Ali Asrorovich; Jarad, Fahd
In this article, an analytical and numerical analysis of a computer virus epidemic model is presented. To more thoroughly examine the dynamics of the virus, the classical model is transformed into a fractional order model. The Caputo differential operator is applied to achieve this. The Jacobian approach is employed to investigate the model's stability. To investigate the model's numerical solution, a hybridized numerical scheme called the Grunwald Letnikov nonstandard finite difference (GL-NSFD) scheme is created. Some essential characteristics of the population model are scrutinized, including positivity boundedness and scheme stability. The aforementioned features are validated using test cases and computer simulations. The mathematical graphs are all detailed. It is also investigated how the fundamental reproduction number R0 functions in stability analysis and illness dynamics.
Citation Count: Akgül, Ali...et al. (2021). "New applications related to Covid-19", Results in Physics, Vol. 20.
New applications related to Covid-19
(2021) Akgül, Ali; Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur; 56389
Analysis of mathematical models projected for COVID-19 presents in many valuable outputs. We analyze a model of differential equation related to Covid-19 in this paper. We use fractal-fractional derivatives in the proposed model. We analyze the equilibria of the model. We discuss the stability analysis in details. We apply very effective method to obtain the numerical results. We demonstrate our results by the numerical simulations. © 2020 The Author(s)
Citation Count: Shahid, Naveed...et al. (2021). "Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution". AIMS MATHEMATICS. Vol: 6, No: 8, pp. 8294-8314.
Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution
(2021) Shahid, Naveed; Rehman, Muhammad Aziz-ur; Ahmed, Nauman; Baleanu, Dumitru; Iqbal, Muhammad Sajid; Rafiq, Muhammad; 56389
In the recent article, a reaction-advection-diffusion model of the hepatitis-B virus (HBV) is studied. Existence and uniqueness of the optimal solution for the proposed model in function spaces is analyzed. The advection and diffusion terms make the model more generic than the simple model. So, the numerical investigation plays a vital role to understand the behavior of the solutions. To find the existence and uniqueness of the optimal solutions, a closed and convex subset (closed ball) of the Banach space is considered. The explicit estimates regarding the solution of the system for the admissible auxiliary data is computed. On the other hand, for the numerical approximation of the solution, an elegant numerical technique is devised to find the approximate solutions. After constructing the discrete model, some fundamental properties must necessarily be possessed by the proposed numerical scheme. For instance, consistency, stability, and positivity of the solutions. These properties are carefully studied in the current article. To prove the positivity of the proposed scheme, M-matrix theory is used. All the above mentioned properties are verified by sketching the graph via simulations. Furthermore, these plots are helpful to understand the true behavior of the solutions. For this purpose, a fruitful discussion is included about the simulations to justify our results.
Citation Count: Ahmed, Nauman...et al. (2020). "Positive explicit and implicit computational techniques for reaction-diffusion epidemic model of dengue disease dynamics", Advances in Difference Equations, Vol. 2020, No. 1.
Positive explicit and implicit computational techniques for reaction-diffusion epidemic model of dengue disease dynamics
(2020) Ahmed, Nauman; Malik, Muhammad Rafiq; Baleanu, Dumitru; Alshomrani, Ali Saleh; Rehman, Muhammad Aziz-ur; 56389
The aim of this work is to develop a novel explicit unconditionally positivity preserving finite difference (FD) scheme and an implicit positive FD scheme for the numerical solution of dengue epidemic reaction-diffusion model with incubation period of virus. The proposed schemes are unconditionally stable and preserve all the essential properties of the solution of the dengue reaction diffusion model. This proposed FD schemes are unconditionally dynamically consistent with positivity property and converge to the true equilibrium points of dengue epidemic reaction diffusion system. Comparison of the proposed scheme with the well-known existing techniques is also presented. The time efficiency of both the proposed schemes is also compared, with the two widely used techniques.
Citation Count: Iqbal, Zafar...et al. (2020). "Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission", Chaos Solitons & Fractals, Vol. 134.
Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission
(2020) Iqbal, Zafar; Ahmed, Nauman; Baleanu, Dumitru; Adel, Waleed; Rafiq, Muhammad; Rehman, Muhammad Aziz-ur; Alshomrani, Ali Saleh; 56389
In this article, an integer order nonlinear HIV/AIDS infection model is extended to the non-integer nonlinear model. The Grunwald Letnikov nonstandard finite difference scheme is designed to obtain the numerical solutions. Structure preservence is one of the main advantages of this scheme. Reproductive number R-0 is worked out and its key role in disease dynamics and stability of the system is investigated with the following facts, if R-0 < 1 the disease will be diminished and it will persist in the community for R-0 > 1. On the other hand, it is sought out that system is stable when R-0 < 1 and R-0 > 1 implicates that system is locally asymptotically stable. Positivity and boundedness of the scheme is also proved for the generalized system. Two steady states of the system are computed and verified by computer simulations with the help of some suitable test problem. (C) 2020 Elsevier Ltd. All rights reserved.
Citation Count: Jawaz, Muhammad Jawaz...et al. (2020). "Positivity Preserving Technique for the Solution of HIV/AIDS Reaction Diffusion Model With Time Delay", Frontiers in Physics, Vol. 7.
Positivity Preserving Technique for the Solution of HIV/AIDS Reaction Diffusion Model With Time Delay
(2020) Jawaz, Muhammad Jawaz; Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Rehman, Muhammad Aziz-ur; 56389
This study is concerned with finding a numerical solution to the delay epidemic model with diffusion. This is not a simple task as variables involved in the model exhibit some important physical features. We have therefore designed an efficient numerical scheme that preserves the properties acquired by the given system. We also further develop Euler's technique for a delayed epidemic reaction-diffusion model. The proposed numerical technique is also compared with the forward Euler technique, and we observe that the forward Euler technique demonstrates the false behavior at certain step sizes. On the other hand, the proposed technique preserves the true behavior of the continuous system at all step sizes. Furthermore, the effect of the delay factor is discussed graphically by using the proposed technique.
Citation Count: Iqbal, Zafar...et al. (2020). "Structure preserving computational technique for fractional order Schnakenberg model", Computational & Applied Mathematics, Vol. 39, No. 2.
Structure preserving computational technique for fractional order Schnakenberg model
(2020) Iqbal, Zafar; Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; 56389
The current article deals with the analysis and numerical solution of fractional order Schnakenberg (S-B) model. This model is a system of autocatalytic reactions by nature, which arises in many biological systems. This study is aiming at investigating the behavior of natural phenomena with a more realistic and practical approach. The solutions are obtained by applying the Grunwald-Letnikov (G-L) finite difference (FD) and the proposed G-L nonstandard finite difference (NSFD) computational schemes. The proposed formulation is explicit in nature, strongly structure preserving as well as it is independent of the time step size. One very important feature of our proposed scheme is that it preserves the positivity of the solution of continuous fractional order S-B model because the unknown variables involved in this system describe the chemical concentrations of different substances. The comparison of the proposed scheme with G-L FD method reflects the significance of the said method.