Browsing by Author "Rehman, Muhammad Aziz-ur"

Now showing 1 - 12 of 12

Citation - WoS: 20
Citation - Scopus: 26
A finite difference scheme to solve a fractional order epidemic model of computer virus
(Amer inst Mathematical Sciences-aims, 2023) Jarad, Fahd; Rehman, Muhammad Aziz-ur; Imran, Muhammad; Ahmed, Nauman; Fatima, Umbreen; Akgul, Ali; Jarad, Fahd; Matematik
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 - WoS: 28
Citation - Scopus: 32
New applications related to Covid-19
(Elsevier, 2021) Akgul, Ali; Baleanu, Dumitru; Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur; 56389; Matematik
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.
Citation - WoS: 10
Citation - Scopus: 11
Novel numerical analysis for nonlinear advection-reaction-diffusion systems
(de Gruyter Poland Sp Z O O, 2020) Shahid, Naveed; Baleanu, Dumitru; Ahmed, Nauman; Baleanu, Dumitru; Alshomrani, Ali Saleh; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; Malik, Muhammad Rafiq; 56389; Matematik
In this article, a numerical model for a Brusselator advection-reaction-diffusion (BARD) system by using an elegant numerical scheme is developed. The consistency and stability of the proposed scheme is demonstrated. Positivity preserving property of the proposed scheme is also verified. The designed scheme is compared with the two well-known existing classical schemes to validate the certain physical properties of the continuous system. A test problem is also furnished for simulations to support our claim. Prior to computations, the existence and uniqueness of solutions for more generic problems is investigated. In the underlying system, the nonlinearities depend not only on the desired solution but also on the advection term that reflects the pivotal importance of the study.
Citation - WoS: 2
Citation - Scopus: 2
Numerical investigation for the nonlinear model of hepatitis-B virus with the existence of optimal solution
(Amer inst Mathematical Sciences-aims, 2021) Shahid, Naveed; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur; Ahmed, Nauman; Baleanu, Dumitru; Iqbal, Muhammad Sajid; Rafiq, Muhammad; 56389; Matematik
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 - WoS: 11
Citation - Scopus: 14
Numerical study of computer virus reaction diffusion epidemic model
(Tech Science Press, 2021) Fatima, Umbreen; Baleanu, Dumitru; Baleanu, Dumitru; Ahmed, Nauman; Azam, Shumaila; Raza, Ali; Rafiq, Muhammad; Rehman, Muhammad Aziz-ur; 56389; Matematik
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 - WoS: 8
Citation - Scopus: 8
Optimality of solution with numerical investigation for coronavirus epidemic model
(Tech Science Press, 2021) Shahid, Naveed; Baleanu, Dumitru; Baleanu, Dumitru; Ahmed, Nauman; Shaikh, Tahira Sumbal; Raza, Ali; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; 56389; Matematik
The novel coronavirus disease, coined as COVID-19, is a murderous and infectious disease initiated from Wuhan, China. This killer disease has taken a large number of lives around the world and its dynamics could not be controlled so far. In this article, the spatio-temporal compartmental epidemic model of the novel disease with advection and diffusion process is projected and analyzed. To counteract these types of diseases or restrict their spread, mankind depends upon mathematical modeling and medicine to reduce, alleviate, and anticipate the behavior of disease dynamics. The existence and uniqueness of the solution for the proposed system are investigated. Also, the solution to the considered system is made possible in a well-known functions space. For this purpose, a Banach space of function is chosen and the solutions are optimized in the closed and convex subset of the space. The essential explicit estimates for the solutions are investigated for the associated auxiliary data. The numerical solution and its analysis are the crux of this study. Moreover, the consistency, stability, and positivity are the indispensable and core properties of the compartmental models that a numerical design must possess. To this end, a nonstandard finite difference numerical scheme is developed to find the numerical solutions which preserve the structural properties of the continuous system. The M-matrix theory is applied to prove the positivity of the design. The results for the consistency and stability of the design are also presented in this study. The plausibility of the projected scheme is indicated by an appropriate example. Computer simulations are also exhibited to conclude the results.
Citation - WoS: 6
Citation - Scopus: 10
Positive explicit and implicit computational techniques for reaction-diffusion epidemic model of dengue disease dynamics
(Springer, 2020) Ahmed, Nauman; Baleanu, Dumitru; Malik, Muhammad Rafiq; Baleanu, Dumitru; Alshomrani, Ali Saleh; Rehman, Muhammad Aziz-ur; 56389; Matematik
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 - WoS: 51
Citation - Scopus: 62
Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission
(Pergamon-elsevier Science Ltd, 2020) Iqbal, Zafar; Baleanu, Dumitru; Ahmed, Nauman; Baleanu, Dumitru; Adel, Waleed; Rafiq, Muhammad; Rehman, Muhammad Aziz-ur; Alshomrani, Ali Saleh; 56389; Matematik
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 - WoS: 6
Citation - Scopus: 12
Positivity Preserving Technique for the Solution of HIV/AIDS Reaction Diffusion Model With Time Delay
(Frontiers Media Sa, 2020) Jawaz, Muhammad Jawaz; Baleanu, Dumitru; Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Rehman, Muhammad Aziz-ur; 56389; Matematik
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 - WoS: 4
Citation - Scopus: 7
Structure preserving algorithms for mathematical model of auto-catalytic glycolysis chemical reaction and numerical simulations
(Springer Heidelberg, 2020) Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur; Khan, Ilyas; Ali, Mubasher; Nisar, Kottakkaran Sooppy; 56389; Matematik
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.
Citation - WoS: 14
Citation - Scopus: 14
Structure preserving computational technique for fractional order Schnakenberg model
(Springer Heidelberg, 2020) Iqbal, Zafar; Baleanu, Dumitru; Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; 56389; Matematik
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.
Citation - WoS: 1
Citation - Scopus: 1
Structure Preserving Numerical Analysis of Reaction-Diffusion Models
(Wiley, 2022) Ahmed, Nauman; Jarad, Fahd; Rehman, Muhammad Aziz-ur; Adel, Waleed; Jarad, Fahd; Ali, Mubasher; Rafiq, Muhammad; Akgul, Ali; 234808; Matematik
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.