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| 1 | A Reliable Stochastic Numerical Analysis for Typhoid Fever Incorporating With Protection Against Infection显示文摘In this paper,a reliable stochastic numerical analysis for typhoid fever incorporating with protection against infection has been considered.We have compared the solutions of stochastic and deterministic typhoid fever model.It has been shown that the stochastic typhoid fever model is more realistic as compared to the deterministic typhoid fever model.The effect of threshold number T*hold in stochastic typhoid fever model.The proposed framework of the stochastic non-standard finite difference scheme(SNSFD)preserves all dynamical properties like positivity,bounded-ness and dynamical consistency defined by Mickens,R.E.The stochastic numerical simulation of the model showed that increase in protection leads to low disease prevalence in a population. | Muhammad Shoaib Arif Ali Raza Muhammad Rafiq Mairaj Bibi Rabia Fayyaz Mehvish Naz Umer Javed | 2019 | Computers, Materials & Continua2019，，6： | 3 |
| 2 | A Stochastic Numerical Analysis for Computer Virus Model with Vertical Transmission Over the Internet显示文摘We are presenting the numerical analysis for stochastic SLBR model of computer virus over the internet in this manuscript.We are going to present the results of stochastic and deterministic computer virus model.Outcomes of the threshold number C∗hold in stochastic computer virus model.If C∗<1 then in such a condition virus controlled in the computer population while C∗>1 shows virus spread in the computer population.Unfortunately,stochastic numerical techniques fail to cope with large step sizes of time.The suggested structure of the stochastic non-standard finite difference scheme(SNSFD)maintains all diverse characteristics such as dynamical consistency,bounded-ness and positivity as well-defined by Mickens.On this basis,we can suggest a collection of plans for eradicating viruses spreading across the internet effectively. | Muhammad Shoaib Arif Ali Raza Wasfi Shatanawi Muhammad Rafiq Mairaj Bibi | 2019 | Computers, Materials & Continua2019，，9： | 2 |
| 3 | Stochastic Numerical Analysis for Impact of Heavy Alcohol Consumption on Transmission Dynamics of Gonorrhoea Epidemic显示文摘This paper aims to perform a comparison of deterministic and stochastic models.The stochastic modelling is a more realistic way to study the dynamics of gonorrhoea infection as compared to its corresponding deterministic model.Also,the deterministic solution is itself mean of the stochastic solution of the model.For numerical analysis,first,we developed some explicit stochastic methods,but unfortunately,they do not remain consistent in certain situations.Then we proposed an implicitly driven explicit method for stochastic heavy alcohol epidemic model.The proposed method is independent of the choice of parameters and behaves well in all scenarios.So,some theorems and simulations are presented in support of the article. | Kamaleldin Abodayeh Ali Raza Muhammad Shoaib Arif Muhammad Rafiq Mairaj Bibi Muhammad Mohsin | 2020 | Computers, Materials & Continua2020，，3： | 2 |
| 4 | Computational Modeling of Reaction-Diffusion COVID-19 Model Having Isolated Compartment显示文摘Cases of COVID-19 and its variant omicron are raised all across the world.The most lethal form and effect of COVID-19 are the omicron version,which has been reported in tens of thousands of cases daily in numerous nations.Following WHO(World health organization)records on 30 December 2021,the cases of COVID-19 were found to be maximum for which boarding individuals were found 1,524,266,active,recovered,and discharge were found to be 82,402 and 34,258,778,respectively.While there were 160,989 active cases,33,614,434 cured cases,456,386 total deaths,and 605,885,769 total samples tested.So far,1,438,322,742 individuals have been vaccinated.The coronavirus or COVID-19 is inciting panic for several reasons.It is a new virus that has affected the whole world.Scientists have introduced certain ways to prevent the virus.One can lower the danger of infection by reducing the contact rate with other persons.Avoiding crowded places and social events withmany people reduces the chance of one being exposed to the virus.The deadly COVID-19 spreads speedily.It is thought that the upcoming waves of this pandemicwill be evenmore dreadful.Mathematicians have presented severalmathematical models to study the pandemic and predict future dangers.The need of the hour is to restrict the mobility to control the infection from spreading.Moreover,separating affected individuals from healthy people is essential to control the infection.We consider the COVID-19 model in which the population is divided into five compartments.The present model presents the population’s diffusion effects on all susceptible,exposed,infected,isolated,and recovered compartments.The reproductive number,which has a key role in the infectious models,is discussed.The equilibrium points and their stability is presented.For numerical simulations,finite difference(FD)schemes like nonstandard finite difference(NSFD),forward in time central in space(FTCS),and Crank Nicolson(CN)schemes are implemented.Some core characteristics of schemes like stability and consistency are calculated. | Muhammad Shoaib Arif Kamaleldin Abodayeh Asad Ejaz | 2023 | Computer Modeling in Engineering & Sciences2023，，5： | 1 |
| 5 | Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm显示文摘This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices.We choose the geodesic distance between��AHX��XA and P as the cost function,and put forward the Extended Hamiltonian algorithm(EHA)and Natural gradient algorithm(NGA)for the solution.Finally,several numerical experiments give you an idea about the effectiveness of the proposed algorithms.We also show the comparison between these two algorithms EHA and NGA.Obtained results are provided and analyzed graphically.We also conclude that the extended Hamiltonian algorithm has better convergence speed than the natural gradient algorithm,whereas the trajectory of the solution matrix is optimal in case of Natural gradient algorithm(NGA)as compared to Extended Hamiltonian Algorithm(EHA).The aim of this paper is to show that the Extended Hamiltonian algorithm(EHA)has superior convergence properties as compared to Natural gradient algorithm(NGA).Upto the best of author’s knowledge,no approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices is found so far in the literature. | Muhammad Shoaib Arif Mairaj Bibi Adnan Jhangir | 2018 | Computers, Materials & Continua2018，，2： | 1 |
| 6 | Mass Transfer of MHD Nanofluid in Presence of Chemical Reaction on A Permeable Rotating Disk with Convective Boundaries, Using Buongiorno’s Model显示文摘This communiquéis opted to study the flow of nanofluid because of heated disk rotation subjected to the convective boundaries with chemical reaction of first order.Wherein Buongiorno’s model for nanofluids is used due to its wide range of applications and the rotating disk under investigation is permeable.Small magneto Reynolds parameter and boundary layer assumptions are carried out to formulate the problem.The system of nonlinear partial differential equations governing the flow problem is converted into the set of ordinary differential equations by using particular relations known as Von Karman transformations.The complicated set of coupled ordinary differential equations with complicated boundary conditions is set to solve by an analytical technique Homotopy Analysis Method(HAM).Whereby the results obtained by the aforementioned method are provided analytically and analyzed graphically.Also validation of the work is confirmed by providing comparison of previous works in tabular form.Effect of chemical reaction parameter on mass transfer rate is also highlighted tabularly for its increament.Nusselt and Sherwood numbers calculated and compared to the like literature and found in good agreement. | Muhammad Shoaib Arif Yasir Nawaz Mairaj Bibi Zafar Ali | 2018 | Computer Modeling in Engineering & Sciences2018，，7： | 1 |
| 7 | Thermal Analysis of MHD Non-Newtonian Nanofluids over a Porous Media显示文摘In the present research,Tiwari and Das model are used for the impact of a magnetic field on non-Newtonian nanofluid flow in the presence of injection and suction.The PDEs are converted into ordinary differential equations(ODEs)using the similarity method.The obtained ordinary differential equations are solved numerically using shooting method along with RK-4.Part of the present study uses nanoparticles(NPs)like TiO_(2) andAl_(2)O_(3) and sodium carboxymethyl cellulose(CMC/water)is considered as a base fluid(BF).This study is conducted to find the influence of nanoparticles,Prandtl number,and magnetic field on velocity and temperature profile,however,the Nusselt number and coefficient of skin friction parameters are also presented in detail with the variation of nanoparticles and parameters.The obtained results of the present study are presented usingMATLAB.In addition to these,some simulations of partial differential equations are also shown using software for graphing surface plots of velocity profile and streamlines along with surface plots and isothermal contours of the temperature profile. | Asad Ejaz Imran Abbas Yasir Nawaz Muhammad Shoaib Arif Wasfi Shatanawi Javeria Nawaz Abbasi | 2020 | Computer Modeling in Engineering & Sciences2020，，12： | 1 |
| 8 | Essential Features Preserving Dynamics of Stochastic Dengue Model显示文摘Nonlinear stochasticmodelling plays an important character in the different fields of sciences such as environmental,material,engineering,chemistry,physics,biomedical engineering,and many more.In the current study,we studied the computational dynamics of the stochastic dengue model with the real material of the model.Positivity,boundedness,and dynamical consistency are essential features of stochastic modelling.Our focus is to design the computational method which preserves essential features of the model.The stochastic non-standard finite difference technique is most efficient as compared to other techniques used in literature.Analysis and comparison were explored in favour of convergence.Also,we address the comparison between the stochastic and deterministic models. | Wasfi Shatanawi Ali Raza Muhammad Shoaib Arif Muhammad Rafiq Mairaj Bibi Muhammad Mohsin | 2021 | Computer Modeling in Engineering & Sciences2021，，1： | 1 |
| 9 | A Computational Analysis to Burgers Huxley Equation显示文摘The efficiency of solving computationally partial differential equations can be profoundly highlighted by the creation of precise,higher-order compact numerical scheme that results in truly outstanding accuracy at a given cost.The objective of this article is to develop a highly accurate novel algorithm for two dimensional non-linear Burgers Huxley(BH)equations.The proposed compact numerical scheme is found to be free of superiors approximate oscillations across discontinuities,and in a smooth ow region,it efciently obtained a high-order accuracy.In particular,two classes of higherorder compact nite difference schemes are taken into account and compared based on their computational economy.The stability and accuracy show that the schemes are unconditionally stable and accurate up to a two-order in time and to six-order in space.Moreover,algorithms and data tables illustrate the scheme efciency and decisiveness for solving such non-linear coupled system.Efciency is scaled in terms of L_(2) and L_(∞) norms,which validate the approximated results with the corresponding analytical solution.The investigation of the stability requirements of the implicit method applied in the algorithm was carried out.Reasonable agreement was constructed under indistinguishable computational conditions.The proposed methods can be implemented for real-world problems,originating in engineering and science. | Muhammad Saqib Muhammad Shoaib Arif Shahid Hasnain Daoud S.Mashat | 2021 | Computers, Materials & Continua2021，，5： | 0 |
| 10 | Analysis of Pneumonia Model via Efficient Computing Techniques显示文摘Pneumonia is a highly transmissible disease in children.According to the World Health Organization(WHO),the most affected regions include south Asia and sub-Saharan Africa.Worldwide,15%of pediatric deaths can be attributed to pneumonia.Computing techniques have a significant role in science,engineering,and many other fields.In this study,we focused on the efficiency of numerical techniques via computer programs.We studied the dynamics of the pneumonia-like infections of epidemic models using numerical techniques.We discuss two types of analysis:dynamical and numerical.The dynamical analysis included positivity,boundedness,local stability,reproduction number,and equilibria of the model.We also discusswell-known computing techniques including Euler,Runge Kutta,and non-standard finite difference(NSFD)for the model.The non-standard finite difference(NSFD)technique shows convergence to the true equilibrium points of the model for any time step size.However,Euler and Runge Kutta do not work well over large time intervals.Computing techniques are the suitable tool for crosschecking the theoretical analysis of the model. | Kamaledin Abodayeh Ali Raza Muhammad Rafiq Muhammad Shoaib Arif Muhammad Naveed Zunir Zeb Syed Zaheer Abbas Kiran Shahzadi Sana Sarwar Qasim Naveed Badar Ul Zaman Muhammad Mohsin | 2022 | Computers, Materials & Continua2022，，3： | 0 |
| 11 | A Hybrid SIR-Fuzzy Model for Epidemic Dynamics:A Numerical Study显示文摘This study focuses on the urgent requirement for improved accuracy in diseasemodeling by introducing a newcomputational framework called the Hybrid SIR-Fuzzy Model.By integrating the traditional Susceptible-Infectious-Recovered(SIR)modelwith fuzzy logic,ourmethod effectively addresses the complex nature of epidemic dynamics by accurately accounting for uncertainties and imprecisions in both data and model parameters.The main aim of this research is to provide a model for disease transmission using fuzzy theory,which can successfully address uncertainty in mathematical modeling.Our main emphasis is on the imprecise transmission rate parameter,utilizing a three-part description of its membership level.This enhances the representation of disease processes with greater complexity and tackles the difficulties related to quantifying uncertainty in mathematical models.We investigate equilibrium points for three separate scenarios and perform a comprehensive sensitivity analysis,providing insight into the complex correlation betweenmodel parameters and epidemic results.In order to facilitate a quantitative analysis of the fuzzy model,we propose the implementation of a resilient numerical scheme.The convergence study of the scheme demonstrates its trustworthiness,providing a conditionally positive solution,which represents a significant improvement compared to current forward Euler schemes.The numerical findings demonstrate themodel’s effectiveness in accurately representing the dynamics of disease transmission.Significantly,when the mortality coefficient rises,both the susceptible and infected populations decrease,highlighting the model’s sensitivity to important epidemiological factors.Moreover,there is a direct relationship between higher Holling type rate values and a decrease in the number of individuals who are infected,as well as an increase in the number of susceptible individuals.This correlation offers a significant understanding of how many elements affect the consequences of an epidemic.Our objective is to enhance decision-making in public health by providing a thorough quantitative analysis of the Hybrid SIR-Fuzzy Model.Our approach not only tackles the existing constraints in disease modeling,but also paves the way for additional investigation,providing a vital instrument for researchers and policymakers alike. | Muhammad Shoaib Arif Kamaleldin Abodayeh Yasir Nawaz | 2024 | Computer Modeling in Engineering & Sciences2024，139，6： | 0 |
| 12 | A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow Over a Linearly Stretching Surface显示文摘Present contribution is concerned with the construction and application of a numerical method for the fluid flow problem over a linearly stretching surface with the modification of standard Gradient descent Algorithm to solve the resulted difference equation.The flow problem is constructed using continuity,and Navier Stoke equations and these PDEs are further converted into boundary value problem by applying suitable similarity transformations.A central finite difference method is proposed that gives third-order accuracy using three grid points.The stability conditions of the present proposed method using a Gauss-Seidel iterative procedure is found using Von-Neumann stability criteria and order of the finite difference method is proved by applying the Taylor series on the discretised equation.The comparison of the presently modified optimisation algorithm with the Gauss-Seidel iterative method and standard Newton’s method in optimisation is also made.It can be concluded that the presently modified optimisation Algorithm takes a few iterations to converge with a small value of the parameter contained in it compared with the standard descent algorithm that may take millions of iterations to converge.The present modification of the steepest descent method converges faster than Gauss-Seidel method and standard steepest descent method,and it may also overcome the deficiency of singular hessian arise in Newton’s method for some of the cases that may arise in optimisation problem(s). | Yasir Nawaz Muhammad Shoaib Arif Mairaj Bibi Javeria Nawaz Abbasi Umer Javed Amna Nazeer | 2020 | Computers, Materials & Continua2020，，2： | 0 |
| 13 | Numerical Treatment for Stochastic Computer Virus Model显示文摘This writing is an attempt to explain a reliable numerical treatment for stochastic computer virus model.We are comparing the solutions of stochastic and deterministic computer virus models.This paper reveals that a stochastic computer virus paradigm is pragmatic in contrast to the deterministic computer virus model.Outcomes of threshold number C^?hold in stochastic computer virus model.If C^?<1 then in such a condition virus controlled in the computer population while C^?>1 shows virus persists in the computer population.Unfortunately,stochastic numerical methods fail to cope with large step sizes of time.The suggested structure of the stochastic non-standard finite difference scheme(SNSFD)maintains all diverse characteristics such as dynamical consistency,boundedness and positivity as defined by Mickens.The numerical treatment for the stochastic computer virus model manifested that increasing the antivirus ability ultimates small virus dominance in a computer community. | Ali Raza Muhammad Shoaib Arif Muhammad Rafiq Mairaj Bibi Muhammad Naveed Muhammad Usman Iqbal Zubair Butt Hafiza Anum Naseem Javeria Nawaz Abbasi | 2019 | Computer Modeling in Engineering & Sciences2019，，8： | 0 |
| 14 | Numerical Simulations for Stochastic Computer Virus Propagation Model显示文摘We are presenting the numerical simulations for the stochastic computer virus propagation model in this manuscript.We are comparing the solutions of stochastic and deterministic computer virus models.Outcomes of a threshold number R0 hold in stochastic computer virus model.If R_(0)<1 then in such a condition virus controlled in the computer population while R_(0)>1 shows virus rapidly spread in the computer population.Unfortunately,stochastic numerical techniques fail to cope with large step sizes of time.The suggested structure of the stochastic non-standard finite difference technique can never violate the dynamical properties.On this basis,we can suggest a collection of strategies for removing virus’s propagation in the computer population. | Muhammad Shoaib Arif Ali Raza Muhammad Rafiq Mairaj Bibi Javeria Nawaz Abbasi Amna Nazeer Umer Javed | 2020 | Computers, Materials & Continua2020，，1： | 0 |
| 15 | An Effective Numerical Method for the Solution of a Stochastic Coronavirus(2019-nCovid)Pandemic Model显示文摘Nonlinear stochastic modeling plays a significant role in disciplines such as psychology,finance,physical sciences,engineering,econometrics,and biological sciences.Dynamical consistency,positivity,and boundedness are fundamental properties of stochastic modeling.A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation.Well-known explicit methods such as Euler Maruyama,stochastic Euler,and stochastic Runge–Kutta are investigated for the stochastic model.Regrettably,the above essential properties are not restored by existing methods.Hence,there is a need to construct essential properties preserving the computational method.The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model.The comparison of the results of deterministic and stochastic models is also presented.Our proposed efficient computational method well preserves the essential properties of the model.Comparison and convergence analyses of the method are presented. | Wasfi Shatanawi Ali Raza Muhammad Shoaib Arif Kamaledin Abodayeh Muhammad Rafiq Mairaj Bibi | 2021 | Computers, Materials & Continua2021，，2： | 0 |
| 16 | Stability Analysis of Predator-Prey System with Consuming Resource and Disease in Predator Species显示文摘The present study is concerned with formulating a predator-prey eco-epidemiological mathematical model assuming that an infection exists in the predator species.The two classes of predator species(susceptible and infected)compete for the same sources available in the environment with the predation option.It is assumed that the disease does not spread vertically.The proposed model is analyzed for the stability of the coexistence of the predators and prey.The fixed points are carried out,and the coexisting fixed point is studied in detail by constructing the Lyapunov function.The movement of species in search of food or protection in their habitat has a significant influence,examined through diffusion.The ecological influences of self-diffusion on the population density of both species are studied.It is theoretically proved that all the under consideration species can coexist in the same environment.The coexistence fixed point is discussed for both diffusive and non-diffusive cases.Moreover,a numerical scheme is constructed for solving time-dependent partial differential equations.The stability of the scheme is given,and it is applied for solving presently modified eco-epidemiological mathematical model with and without diffusion.The comparison of the constructed scheme with two exiting schemes,Backward in Time and Central in Space(BTCS)and Crank Nicolson,is also given in the form of plots.Finally,we run a computer simulation to determine the effectiveness of the proposed numerical scheme.For readers’convenience,a computational code for the proposed discrete model scheme may be made available upon request. | Asad Ejaz Yasir Nawaz Muhammad Shoaib Arif Daoud S.Mashat Kamaleldin Abodayeh | 2022 | Computer Modeling in Engineering & Sciences2022，，8： | 0 |
| 17 | A reliable numerical analysis for stochastic gonorrhea epidemic model with treatment effect显示文摘The phenomena of disease spread are unpredictable in nature due to random mixing of individuals in a population.It is of more significance to include this randomness while modeling infectious diseases.Modeling epidemics including tlieir stochastic behavior could be a more realistic approach in many situations.In thus paper,a stochastic gon orrhea epidemic model with treatment effect has been analyzed numerically.Numerical solution of stochastic model is presented in comparison with its deterministic part.The dynamics of the gonorrhea disease is governed by a threshold quantity Ai called basic reproductive number.If.A1<1.then disease eventually dies out while A1>1 shows the persistence of disease in population.The standard numerical schemes like Euler Maruyaina.stochastic Euler and stochastic Rurige Kutta are highly dependent on step size and do not behave well in certain scenarios.A competitive non-standard finite dif ference numerical scheme in stochastic setting is proposed,which is independent of step size and remains consistent with the corresponding deterministic model. | Ali Raza Muhammad Shoaib Arif Muhammad Ualiq | 2019 | International Journal of Biomathematics2019，12，6： | 0 |
| 18 | Numerical Study for Magnetohydrodynamic(MHD)Unsteady Maxwell Nanofluid Flow Impinging on Heated Stretching Sheet显示文摘The numerous applications of Maxwell Nanofluid Stagnation Point Flow,such as those in production industries,the processing of polymers,compression,power generation,lubrication systems,food manufacturing and air conditioning,among other applications,require further research into the effects of various parameters on flow phenomena.In this paper,a study has been carried out for the heat andmass transfer of Maxwell nanofluid flow over the heated stretching sheet.A mathematical model with constitutive expressions is constructed in partial differential equations(PDEs)through obligatory basic conservation laws.A series of transformations are then used to take the system into an ordinary differential equation(ODE).The boundary conditions(BCs)are also treated similarly for transforming into first-order ordinary differential equations(ODEs).Then these ODEs are computed by using the Shooting Method.The effect of factors on the skin friction coefficient,the local Nusselt number,and the local Sherwood number are explored,and the results are displayed graphically.The obtained results demonstrate that by increasing the values of the Maxwell and slip velocity parameters,velocity deescalates.For investigators tasked with addressing unresolved difficulties in the realm of enclosures used in industry and engineering,we thought this work would serve as a guide. | Muhammad Shoaib Arif Muhammad Jhangir Yasir Nawaz Imran Abbas Kamaleldin Abodayeh Asad Ejaz | 2022 | Computer Modeling in Engineering & Sciences2022，，11： | 0 |
| 19 | Hypofractionated chemoradiation in locally advanced squamous cell carcinoma of the buccal mucosa显示文摘 | Arif Jamshed Raza Hussain Muhammad A Ilyas Shoaib Ahmed Hassan Iqbal | 2008 | 中国口腔颌面外科杂志2008，6，B05： | 0 |
| 20 | Numerical Analysis of Stochastic Vector Borne Plant Disease Model显示文摘We are associating the solutions of stochastic and deterministic vector borne plant disease model in this manuscript.The dynamics of plant model depends upon threshold number P^(∗).If P^(∗)<1 then condition helpful to eradicate the disease in plants while P^(∗)>1 explains the persistence of disease.Inappropriately,standard numerical systems do not behave well in certain scenarios.We have been proposed a structure preserving stochastic non-standard finite difference system to analyze the behavior of model.This system is dynamical consistent,positive and bounded as defined by Mickens. | Kamaledin Abodayeh Ali Raza Muhammad Shoaib Arif Muhammad Rafiq Mairaj Bibi Rabia Fayyaz | 2020 | Computers, Materials & Continua2020，，4： | 0 |
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