Browsing by Author "Ahmed, Nauman"

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Citation - WoS: 12
Citation - Scopus: 13
A Novel Time Efficient Structure-Preserving Splitting Method for the Solution of Two-Dimensional Reaction-Diffusion Systems
(Springer, 2020) Korkmaz, Alper; Rafiq, M.; Baleanu, Dumitru; Alshomrani, Ali Saleh; Rehman, M. A.; Iqbal, M. S.; Ahmed, Nauman
In this article, the first part is concerned with the important questions related to the existence and uniqueness of solutions for nonlinear reaction-diffusion systems. Secondly, an efficient positivity-preserving operator splitting nonstandard finite difference scheme (NSFD) is designed for such a class of systems. The presented formulation is unconditionally stable as well as implicit in nature and even time efficient. The proposed NSFD operator splitting technique also preserves all the important properties possessed by continuous systems like positivity, convergence to the fixed points of the system, and boundedness. The proposed algorithm is implicit in nature but more efficient in time than the extensively used Euler method.
Citation - WoS: 9
Citation - Scopus: 9
Positivity Preserving Computational Techniques for Nonlinear Autocatalytic Chemical Reaction Model
(Editura Acad Romane, 2020) Ahmed, Nauman; Baleanu, Dumitru; Baleanu, Dumitru; Korkmaz, Alper; Rafiq, Muhammad; Rehman, Muhammad Aziz-Ur; Ali, Mubasher; Matematik
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.
Numerical Analysis for the Effect of Irresponsible Immigrants on Hiv/Aids Dynamics
(Tech Science Press, 2023) Baleanu, Dumitru; Rafiq, Muhammad; Awrejcewicz, Jan; Ahmed, Nauman; Raza, Ali; Ahmad, Muhammad Ozair; Ali, Muhammad Tariq
The human immunodeficiency viruses are two species of Lentivirus that infect humans. Over time, they cause acquired immunodeficiency syndrome, a condition in which progressive immune system failure allows life-threatening opportunistic infections and cancers to thrive. Human immunodeficiency virus infection came from a type of chimpanzee in Central Africa. Studies show that immunodeficiency viruses may have jumped from chimpanzees to humans as far back as the late 1800s. Over decades, human immunodeficiency viruses slowly spread across Africa and later into other parts of the world. The Susceptible-Infected-Recovered (SIR) models are significant in studying disease dynamics. In this paper, we have studied the effect of irresponsible immigrants on HIV/AIDS dynamics by formulating and considering different methods. Euler, Runge Kutta, and a Non-standard finite difference (NSFD) method are developed for the same problem. Numerical experiments are performed at disease-free and endemic equilibria points at different time step sizes 'h'. The results reveal that, unlike Euler and Runge Kutta, which fail for large time step sizes, the proposed Non-standard finite difference (NSFD) method gives a convergence solution for any time step size. Our proposed numerical method is bounded, dynamically consistent, and preserves the positivity of the continuous solution, which are essential requirements when modeling a prevalent disease.
Citation - WoS: 16
Citation - Scopus: 17
Numerical Analysis of the Susceptible Exposed Infected Quarantined and Vaccinated (Seiqv) Reaction-Diffusion Epidemic Model
(Frontiers Media Sa, 2020) Fatima, Mehreen; Baleanu, Dumitru; Nisar, Kottakkaran Sooppy; Khan, Ilyas; Rafiq, Muhammad; Ahmad, Muhammad Ozair; Ahmed, Nauman
In this paper, two structure-preserving nonstandard finite difference (NSFD) operator splitting schemes are designed for the solution of reaction diffusion epidemic models. The proposed schemes preserve all the essential properties possessed by the continuous systems. These schemes are applied on a diffusive SEIQV epidemic model with a saturated incidence rate to validate the results. Furthermore, the stability of the continuous system is proved, and the bifurcation value is evaluated. A comparison is also made with the existing operator splitting numerical scheme. Simulations are also performed for numerical experiments.
Citation - WoS: 11
Citation - Scopus: 13
Novel Numerical Analysis for Nonlinear Advection-Reaction Systems
(de Gruyter Poland Sp Z O O, 2020) Ahmed, Nauman; Baleanu, Dumitru; Alshomrani, Ali Saleh; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; Malik, Muhammad Rafiq; Shahid, Naveed
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: 8
Citation - Scopus: 12
Positive Explicit and Implicit Computational Techniques for Reaction-Diffusion Epidemic Model of Dengue Disease Dynamics
(Springer, 2020) Malik, Muhammad Rafiq; Baleanu, Dumitru; Alshomrani, Ali Saleh; Rehman, Muhammad Aziz-ur; Ahmed, Nauman
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: 16
Citation - Scopus: 17
Structure Preserving Computational Technique for Fractional Order Schnakenberg Model
(Springer Heidelberg, 2020) Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; Iqbal, Zafar
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: 11
Citation - Scopus: 12
Investigation of Electromagnetic Wave Structures for a Coupled Model in Anti-Ferromagnetic Spin Ladder Medium
(Frontiers Media Sa, 2020) Yousaf, Umair; Ahmed, Nauman; Rizvi, Syed Tahir Raza; Iqbal, Muhammad Sajid; Baleanu, Dumitru; Younis, Muhammad
The article studies the extraction of electromagnetic wave structures in a spin ladder anti-ferromagnetic medium with a coupled generalized non-linear Schrodinger model. The direct algebraic technique is used to extract the wave solutions. The solutions are obtained in the form of dark, singular, kink, and dark-singular under different constraint conditions. Moreover, the dynamic behavior of the structures have depicted in 3D graphs and their corresponding counterplots. The results are helpful for the understanding of wave propagation study and are also vital for numerical and experimental verifications in the field of electromagnetic wave theory.
Citation - WoS: 1
Citation - Scopus: 1
Structure Preserving Numerical Analysis of Reaction-Diffusion Models
(Wiley, 2022) Rehman, Muhammad Aziz-ur; Adel, Waleed; Jarad, Fahd; Ali, Mubasher; Rafiq, Muhammad; Akgul, Ali; Ahmed, Nauman
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.
Citation - WoS: 31
Citation - Scopus: 29
Dynamical Behavior and Sensitivity Analysis of a Delayed Coronavirus Epidemic Model
(Tech Science Press, 2020) Baleanu, Dumitru; Rafiq, Muhammad; Raza, Ali; Soori, Atif Hassan; Ahmed, Nauman; Naveed, Muhammad
Mathematical delay modelling has a significant role in the different disciplines such as behavioural, social, physical, biological engineering, and bio-mathematical sciences. The present work describes mathematical formulation for the transmission mechanism of a novel coronavirus (COVID-19). Due to the unavailability of vaccines for the coronavirus worldwide, delay factors such as social distance, quarantine, travel restrictions, extended holidays, hospitalization, and isolation have contributed to controlling the coronavirus epidemic. We have analysed the reproduction number and its sensitivity to parameters. If, Rcovid 1 then this situation will help to eradicate the disease and if, Rcovid 1 the virus will spread rapidly in the human beings. Well-known theorems such as Routh Hurwitz criteria and Lasalle invariance principle have presented for stability. The local and global stabilizes for both equilibria of the model have also been presented. Also, we have analysed the effect of delay reason on the reproduction number. In the last, some very useful numerical consequences have presented in support of hypothetical analysis.
Citation - WoS: 1
Citation - Scopus: 1
Optimization of Coronavirus Pandemic Model Through Artificial Intelligence
(Tech Science Press, 2023) Nasir, Arooj; Baleanu, Dumitru; Raza, Ali; Cheema, Tahir Nawaz; Ahmed, Nauman; Mahmoud, Emad E.; Alqarni, Manal. M.
Artificial intelligence is demonstrated by machines, unlike the natural intelligence displayed by animals, including humans. Artificial intelligence research has been defined as the field of study of intelligent agents, which refers to any system that perceives its environment and takes actions that maximize its chance of achieving its goals. The techniques of intelligent computing solve many applications of mathematical modeling. The research work was designed via a particular method of artificial neural networks to solve the mathematical model of coronavirus. The representation of the mathematical model is made via systems of nonlinear ordinary differential equations. These differential equations are established by collecting the susceptible, the exposed, the symptomatic, super spreaders, infection with asymptomatic, hospitalized, recovery, and fatality classes. The generation of the coronavirus model's dataset is exploited by the strength of the explicit Runge Kutta method for different countries like India, Pakistan, Italy, and many more. The generated dataset is approximately used for training, validation, and testing processes for each cyclic update in Bayesian Regularization Backpropagation for the numerical treatment of the dynamics of the desired model. The performance and effectiveness of the designed methodology are checked through mean squared error, error histograms, numerical solutions, absolute error, and regression analysis.
Citation - WoS: 13
Citation - Scopus: 12
Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model
(Mdpi, 2022) Macias-Diaz, Jorge E.; Raza, Ali; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Zafar; Ahmad, Muhammad Ozair; Ahmed, Nauman
Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R-0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R-0 < 1. A similar result is obtained for the endemic equilibrium when R-0 > 1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge-Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.
Citation - WoS: 8
Citation - Scopus: 8
Optimality of Solution With Numerical Investigation for Coronavirus Epidemic Model
(Tech Science Press, 2021) Baleanu, Dumitru; Ahmed, Nauman; Shaikh, Tahira Sumbal; Raza, Ali; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; Shahid, Naveed
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: 15
Design and Numerical Analysis of Fuzzy Nonstandard Computational Methods for the Solution of Rumor Based Fuzzy Epidemic Model
(Elsevier, 2022) Rafiq, Muhammad; Ahmed, Nauman; Baleanu, Dumitru; Raza, Ali; Ahmad, Muhammad Ozair; Iqbal, Muhammad; Dayan, Fazal
This model extends the classical epidemic model for cyber consumerism by introducing fuzziness to the model. Fuzziness arises due to insufficient knowledge, experimental errors, operating conditions and parameters that provide inaccurate information. The concepts of confused, escapers and recovered consumers are uncertain due to the different degrees of confusion, escaping and recovery among the individuals of the cyber consumers. The differences can arise, when the cyber consumers under the consideration having distinct habits, customs and different age groups have different degrees of resistance, etc. The chance of transmission of rumors and recovery rates are considered as fuzzy numbers. A rumor-free and two rumor existing-endemic equilibrium points have been derived for the studied model. The model is then solved numerically with fuzzy forward Euler and fuzzy nonstandard finite difference (FNSFD) methods respectively. The numerical and simulation results show that the proposed FNSFD technique is an efficient and reliable tool to deal with such type of dynamical system. (c) 2022 Elsevier B.V. All rights reserved.
Citation - WoS: 24
Citation - Scopus: 23
Spatio-Temporal Numerical Modeling of Auto-Catalytic Brusselator Model
(Editura Acad Romane, 2019) Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-Ur; Matematik
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.
Citation - WoS: 28
Citation - Scopus: 32
New Applications Related To Covid-19
(Elsevier, 2021) Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur; Akgul, Ali
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: 18
Citation - Scopus: 19
Construction and Numerical Analysis of a Fuzzy Non-Standard Computational Method for the Solution of an Seiqr Model of Covid-19 Dynamics
(Amer inst Mathematical Sciences-aims, 2022) Ahmed, Nauman; Rafiq, Muhammad; Akgul, Ali; Raza, Ali; Ahmad, Muhammad Ozair; Jarad, Fahd; Dayan, Fazal
This current work presents an SEIQR model with fuzzy parameters. The use of fuzzy theory helps us to solve the problems of quantifying uncertainty in the mathematical modeling of diseases. The fuzzy reproduction number and fuzzy equilibrium points have been derived focusing on a model in a specific group of people having a triangular membership function. Moreover, a fuzzy non-standard finite difference (FNSFD) method for the model is developed. The stability of the proposed method is discussed in a fuzzy sense. A numerical verification for the proposed model is presented. The developed FNSFD scheme is a reliable method and preserves all the essential features of a continuous dynamical system.
Citation - WoS: 4
Citation - Scopus: 4
Computational Analysis for Computer Network Model With Fuzziness
(Tech Science Press, 2023) Baleanu, Dumitru; Dayan, Fazal; Ullah, Sami; Ahmed, Nauman; Rafiq, Muhammad; Raza, Ali; Alfwzan, Wafa F.
A susceptible, exposed, infectious, quarantined and recovered (SEIQR) model with fuzzy parameters is studied in this work. Fuzziness in the model arises due to the different degrees of susceptibility, exposure, infectivity, quarantine and recovery among the computers under consideration due to the different sizes, models, spare parts, the surrounding environments of these PCs and many other factors like the resistance capacity of the individual PC against the virus, etc. Each individual PC has a different degree of infectivity and resistance against infection. In this scenario, the fuzzy model has richer dynamics than its classical counterpart in epidemiology. The reproduction number of the developed model is studied and the equilibrium analysis is performed. Two different techniques are employed to solve the model numerically. Numerical simulations are performed and the obtained results are compared. Positivity and convergence are maintained by the suggested technique which are the main features of the epidemic models.
Citation - WoS: 6
Citation - Scopus: 11
Modeling of Computer Virus Propagation With Fuzzy Parameters
(Tech Science Press, 2023) Ahmed, Nauman; Baleanu, Dumitru; Fatima, Umbreen; Dayan, Fazal; Rafiq, Muhammad; Mahmoud, Emad E.; Alhebshi, Reemah M.
Typically, a computer has infectivity as soon as it is infected. It is a reality that no antivirus programming can identify and eliminate all kinds of viruses, suggesting that infections would persevere on the Internet. To understand the dynamics of the virus propagation in a better way, a computer virus spread model with fuzzy parameters is presented in this work. It is assumed that all infected computers do not have the same contribution to the virus transmission process and each computer has a different degree of infectivity, which depends on the quantity of virus. Considering this, the parameters beta and gamma being functions of the computer virus load, are considered fuzzy numbers. Using fuzzy theory helps us understand the spread of computer viruses more realistically as these parameters have fixed values in classical models. The essential features of the model, like reproduction number and equilibrium analysis, are discussed in fuzzy senses. Moreover, with fuzziness, two numerical methods, the forward Euler technique, and a nonstandard finite difference (NSFD) scheme, respectively, are developed and analyzed. In the evidence of the numerical simulations, the proposed NSFD method preserves the main features of the dynamic system. It can be considered a reliable tool to predict such types of solutions.
Citation - WoS: 20
Citation - Scopus: 21
Stability Analysis and Numerical Simulations of Spatiotemporal Hiv Cd4+t Cell Model With Drug Therapy
(Amer inst Physics, 2020) Elsonbaty, Amr; Adel, Waleed; Baleanu, Dumitru; Rafiq, Muhammad; Ahmed, Nauman
In this study, an extended spatiotemporal model of a human immunodeficiency virus (HIV) CD4+ T cell with a drug therapy effect is proposed for the numerical investigation. The stability analysis of equilibrium points is carried out for temporal and spatiotemporal cases where stability regions in the space of parameters for each case are acquired. Three numerical techniques are used for the numerical simulations of the proposed HIV reaction-diffusion system. These techniques are the backward Euler, Crank-Nicolson, and a proposed structure preserving an implicit technique. The proposed numerical method sustains all the important characteristics of the proposed HIV model such as positivity of the solution and stability of equilibria, whereas the other two methods have failed to do so. We also prove that the proposed technique is positive, consistent, and Von Neumann stable. The effect of different values for the parameters is investigated through numerical simulations by using the proposed method. The stability of the proposed model of the HIV CD4+ T cell with the drug therapy effect is also analyzed.