In this research article, we handle the susceptible infected-recovered (SIR) model of the dengue ... more In this research article, we handle the susceptible infected-recovered (SIR) model of the dengue fever epidemic under Caputo Fabrizio fractional derivative. The dengue fever disease is a complicated disease because of the connection it creates between humans and mosquitoes. This encouraged scientists to understand the various factors that influence the recurrence of dengue fever. A new technique called the Laplace Optimized Decomposition (LODM) is used to solve this model numerically and compared with the 4 th order Runge-Kutta Method (RKM). The solution in the proposed method is in the form of a convergent series with easily computable components. We present the solution via graphs and hence give some remarks about the nature of the solutions.
In this paper, we applied an extension of optimal homotopy asymptotic method (EOHAM) for the appr... more In this paper, we applied an extension of optimal homotopy asymptotic method (EOHAM) for the approximate solution of coupled partial differential equations (PDEs). The obtained results are compared with other results for its efficiency. The order of convergence and residuals are plotted.
In this article New Iterative Method (NIM) is tested upon time fractional coupled ITO system. The... more In this article New Iterative Method (NIM) is tested upon time fractional coupled ITO system. The results obtained by the proposed method are compared with that of Homotopy Perturbation Method (HPM). It is shown that the proposed method is accurate for strongly nonlinear fractional coupled system of PDEs.
The Journal of Nonlinear Sciences and Applications, Oct 31, 2017
The main purpose of this research paper is to develop some sufficient conditions for the existenc... more The main purpose of this research paper is to develop some sufficient conditions for the existence of solution of a nonlinear problem of implicit fractional differential equations (IFDEs) with boundary conditions, using prior estimate method. The distinction of the method applied here is, it does not require compactness of the operator. This idea is the result of motivation from the book of O'Regan and other [R.
We investigate, through a fractional mathematical model, the effects of physical distance on the ... more We investigate, through a fractional mathematical model, the effects of physical distance on the SARS-CoV-2 virus transmission. Two controls are considered in our model for eradication of the spread of COVID-19: media education, through campaigns explaining the importance of social distancing, use of face masks, etc., towards all population, while the second one is quarantine social isolation of the exposed individuals. A general fractional order optimal control problem, and associated optimality conditions of Pontryagin type, are discussed, with the goal to minimize the number of susceptible and infected while maximizing the number of recovered. The extremals are then numerically obtained.
COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, res... more COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, researchers have been concentrating their efforts on understanding the complex dynamics of this widespread illness. Mathematics plays a big role in understanding the mechanism of the spread of this disease by modeling it and trying to find approximate solutions. In this study, we implement a new technique for an approximation of the analytic series solution called the multistep Laplace optimized decomposition method for solving fractional nonlinear systems of ordinary differential equations. The proposed method is a combination of the multistep method, the Laplace transform, and the optimized decomposition method. To show the ability and effectiveness of this method, we chose the COVID-19 model to apply the proposed technique to it. To develop the model, the Caputo-type fractional-order derivative is employed. The suggested algorithm efficacy is assessed using the fourth-order Runge-Kutta m...
The aim of this paper is to extend the application of the reproducing kernel Hilbert space method... more The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] \bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method
The dynamical Brusselator reaction–diffusion system of time-fractional is used to describe chemic... more The dynamical Brusselator reaction–diffusion system of time-fractional is used to describe chemical models and chemical processes with nonlinear oscillation. In this study, the Laplace optimized decomposition scheme is proposed for approximating solutions of three applications of the two-dimensional (2D) reaction–diffusion Brusselator model with the noninteger derivative proposed in the Caputo approach. Complete descriptions of the scheme and solution steps are utilized and mentioned. By applying the procedures of the Laplace inversion operator and truncating the optimized series, the approximate solutions are drawn, tabulated and sketched. Numerical results show the efficiency, reliability and accuracy of the technique for the nonlinear systems of partial differential equations of noninteger-different order derivatives. Finally, focused notes and futures planning works are mentioned with the most-used references.
In this work, we present some numerical results about variable order fractional differential equa... more In this work, we present some numerical results about variable order fractional differential equations (VOFDEs). For the said numerical analysis, we use Bernstein polynomials (BPs) with non-orthogonal basis. The method we use does not need discretization and neither collocation. Hence omitting the said two operations sufficient memory and time can be saved. We establish operational matrices for variable order integration and differentiation which convert the consider problem to some algebraic type matrix equations. The obtained matrix equations are then solved by Matlab 13 to get the required numerical solution for the considered problem. Pertinent examples are provided along with graphical illustration and error analysis to validate the results. Further some theoretical results for time complexity are also discussed.
In this paper, the reproducing kernel Hilbert space method (RKHSM) is applied to find approximate... more In this paper, the reproducing kernel Hilbert space method (RKHSM) is applied to find approximate solution of fractional delay integro differential equations satisfying initial conditions. The fractional derivatives are considered in the Caputo sense. Numerical examples are provided to demonstrate the efficiency and accuracy of the presented method. It is shown that RKHSM gives excellent results when applied to both fractional linear and nonlinear delay integro differential equations. AMS subject classification:
In this paper, we consider a fractional mathematical model describing the co-infection of HBV and... more In this paper, we consider a fractional mathematical model describing the co-infection of HBV and HCV under the non-singular Mittag-Leffler derivative. We also investigate the qualitative analysis for at least one solution and a unique solution by applying the approach fixed point theory. For an approximate solution, the technique of the iterative fractional order Adams–Bashforth scheme has been implemented. The simulation for the proposed scheme has been drawn at various fractional order values lying between (0,1) and integer-order of 1 via using Matlab. All the compartments have shown convergence and stability with time. A detailed comparative result has been given by the different fractional orders, which showed that the stability was achieved more rapidly at low orders.
In this paper, we are applying a novel analytical hybrid method to find the solution of a fuzzy V... more In this paper, we are applying a novel analytical hybrid method to find the solution of a fuzzy Volterra Abel’s integral equation of the second kind. The fuzzy number is used in its parametric form under which the fuzzy Volterra Abel’s integral equation will be converted into a system of integral equations as in a crisp case. Moreover, to solve the general fuzzy Volterra integral equation with Abel’s type kernel, and to show that the proposed method is efficient, a few accurate and simple examples are given for the demonstration of our results.
This paper investigates a new model on coronavirus-19 disease (COVID-19) with three compartments ... more This paper investigates a new model on coronavirus-19 disease (COVID-19) with three compartments including susceptible, infected, and recovered class under Mittag-Leffler type derivative. The mentioned derivative has been introduced by Atangana, Baleanu, and Caputo abbreviated as$(\mathcal{ABC})$(ABC). Upon utilizing fixed point theory, we first prove the existence of at least one solution for the considered model and its uniqueness. Also, some results about stability of Ulam–Hyers type are also established. By applying a numerical technique called fractional Adams–Bashforth (AB) method, we develop a scheme for the approximate solutions to the considered model. Using some real available data, we perform the concerned numerical simulation corresponding to different values of fractional order.
This manuscript is concerned to the existence and stability of HIV/AIDS infection model with frac... more This manuscript is concerned to the existence and stability of HIV/AIDS infection model with fractional order derivative. The corresponding derivative is taken in Caputo-Fabrizio sense, which is a new approach for such type of biological models. With the help of Sumudu transform, some new results are handled. Further for the corresponding results, existence theory and uniqueness for the equilibrium solution are provided via using nonlinear functional analysis and fixed point theory due to Banach.
International Journal of Applied and Computational Mathematics, 2020
In this research work, we considered the numerical solution of an essential fractional dynamical ... more In this research work, we considered the numerical solution of an essential fractional dynamical model of HIV infection of CD4 + T cells, by using a relatively new technique, Chebyshev Wavelet Method. The numerical simulations are performed for integer-order as well as for fractional-order of HIV models. In this research work, the fractional derivatives are defined by Caputo operator. The proposed method is implemented to solve both fractional and integer-order of HIV models. The result obtained by the suggested method are compare with Müntz-Legendre Polynomials, Laplace Transform with Padé Approximation, Developed Bessel Collocation Method, Laplace Decomposition Method and Runge-Kutta fourth-order (RK4) method. The comparison has verified the greatest accuracy then other existing method.
In this paper, optimal homotopy asymptotic method has been extended to seek out the exact solutio... more In this paper, optimal homotopy asymptotic method has been extended to seek out the exact solution of fractional generalized biological population models. The time fractional derivatives are described in the Caputo sense. It optimal homotopy asymptotic method is a new approach for fractional models. The proposed approach presents a procedure by that we have transferred the model to a series of simpler problems which are solvable by hand work applying the Riemann-Liouville fractional integral operator and obtained exact solution of fractional the generalized biological population by adding the solutions of first three simple problems of the series of simpler problems. The new approach provides exact solution in the way of smoothly convergent series.
In this research article, we handle the susceptible infected-recovered (SIR) model of the dengue ... more In this research article, we handle the susceptible infected-recovered (SIR) model of the dengue fever epidemic under Caputo Fabrizio fractional derivative. The dengue fever disease is a complicated disease because of the connection it creates between humans and mosquitoes. This encouraged scientists to understand the various factors that influence the recurrence of dengue fever. A new technique called the Laplace Optimized Decomposition (LODM) is used to solve this model numerically and compared with the 4 th order Runge-Kutta Method (RKM). The solution in the proposed method is in the form of a convergent series with easily computable components. We present the solution via graphs and hence give some remarks about the nature of the solutions.
In this paper, we applied an extension of optimal homotopy asymptotic method (EOHAM) for the appr... more In this paper, we applied an extension of optimal homotopy asymptotic method (EOHAM) for the approximate solution of coupled partial differential equations (PDEs). The obtained results are compared with other results for its efficiency. The order of convergence and residuals are plotted.
In this article New Iterative Method (NIM) is tested upon time fractional coupled ITO system. The... more In this article New Iterative Method (NIM) is tested upon time fractional coupled ITO system. The results obtained by the proposed method are compared with that of Homotopy Perturbation Method (HPM). It is shown that the proposed method is accurate for strongly nonlinear fractional coupled system of PDEs.
The Journal of Nonlinear Sciences and Applications, Oct 31, 2017
The main purpose of this research paper is to develop some sufficient conditions for the existenc... more The main purpose of this research paper is to develop some sufficient conditions for the existence of solution of a nonlinear problem of implicit fractional differential equations (IFDEs) with boundary conditions, using prior estimate method. The distinction of the method applied here is, it does not require compactness of the operator. This idea is the result of motivation from the book of O'Regan and other [R.
We investigate, through a fractional mathematical model, the effects of physical distance on the ... more We investigate, through a fractional mathematical model, the effects of physical distance on the SARS-CoV-2 virus transmission. Two controls are considered in our model for eradication of the spread of COVID-19: media education, through campaigns explaining the importance of social distancing, use of face masks, etc., towards all population, while the second one is quarantine social isolation of the exposed individuals. A general fractional order optimal control problem, and associated optimality conditions of Pontryagin type, are discussed, with the goal to minimize the number of susceptible and infected while maximizing the number of recovered. The extremals are then numerically obtained.
COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, res... more COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, researchers have been concentrating their efforts on understanding the complex dynamics of this widespread illness. Mathematics plays a big role in understanding the mechanism of the spread of this disease by modeling it and trying to find approximate solutions. In this study, we implement a new technique for an approximation of the analytic series solution called the multistep Laplace optimized decomposition method for solving fractional nonlinear systems of ordinary differential equations. The proposed method is a combination of the multistep method, the Laplace transform, and the optimized decomposition method. To show the ability and effectiveness of this method, we chose the COVID-19 model to apply the proposed technique to it. To develop the model, the Caputo-type fractional-order derivative is employed. The suggested algorithm efficacy is assessed using the fourth-order Runge-Kutta m...
The aim of this paper is to extend the application of the reproducing kernel Hilbert space method... more The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] \bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method
The dynamical Brusselator reaction–diffusion system of time-fractional is used to describe chemic... more The dynamical Brusselator reaction–diffusion system of time-fractional is used to describe chemical models and chemical processes with nonlinear oscillation. In this study, the Laplace optimized decomposition scheme is proposed for approximating solutions of three applications of the two-dimensional (2D) reaction–diffusion Brusselator model with the noninteger derivative proposed in the Caputo approach. Complete descriptions of the scheme and solution steps are utilized and mentioned. By applying the procedures of the Laplace inversion operator and truncating the optimized series, the approximate solutions are drawn, tabulated and sketched. Numerical results show the efficiency, reliability and accuracy of the technique for the nonlinear systems of partial differential equations of noninteger-different order derivatives. Finally, focused notes and futures planning works are mentioned with the most-used references.
In this work, we present some numerical results about variable order fractional differential equa... more In this work, we present some numerical results about variable order fractional differential equations (VOFDEs). For the said numerical analysis, we use Bernstein polynomials (BPs) with non-orthogonal basis. The method we use does not need discretization and neither collocation. Hence omitting the said two operations sufficient memory and time can be saved. We establish operational matrices for variable order integration and differentiation which convert the consider problem to some algebraic type matrix equations. The obtained matrix equations are then solved by Matlab 13 to get the required numerical solution for the considered problem. Pertinent examples are provided along with graphical illustration and error analysis to validate the results. Further some theoretical results for time complexity are also discussed.
In this paper, the reproducing kernel Hilbert space method (RKHSM) is applied to find approximate... more In this paper, the reproducing kernel Hilbert space method (RKHSM) is applied to find approximate solution of fractional delay integro differential equations satisfying initial conditions. The fractional derivatives are considered in the Caputo sense. Numerical examples are provided to demonstrate the efficiency and accuracy of the presented method. It is shown that RKHSM gives excellent results when applied to both fractional linear and nonlinear delay integro differential equations. AMS subject classification:
In this paper, we consider a fractional mathematical model describing the co-infection of HBV and... more In this paper, we consider a fractional mathematical model describing the co-infection of HBV and HCV under the non-singular Mittag-Leffler derivative. We also investigate the qualitative analysis for at least one solution and a unique solution by applying the approach fixed point theory. For an approximate solution, the technique of the iterative fractional order Adams–Bashforth scheme has been implemented. The simulation for the proposed scheme has been drawn at various fractional order values lying between (0,1) and integer-order of 1 via using Matlab. All the compartments have shown convergence and stability with time. A detailed comparative result has been given by the different fractional orders, which showed that the stability was achieved more rapidly at low orders.
In this paper, we are applying a novel analytical hybrid method to find the solution of a fuzzy V... more In this paper, we are applying a novel analytical hybrid method to find the solution of a fuzzy Volterra Abel’s integral equation of the second kind. The fuzzy number is used in its parametric form under which the fuzzy Volterra Abel’s integral equation will be converted into a system of integral equations as in a crisp case. Moreover, to solve the general fuzzy Volterra integral equation with Abel’s type kernel, and to show that the proposed method is efficient, a few accurate and simple examples are given for the demonstration of our results.
This paper investigates a new model on coronavirus-19 disease (COVID-19) with three compartments ... more This paper investigates a new model on coronavirus-19 disease (COVID-19) with three compartments including susceptible, infected, and recovered class under Mittag-Leffler type derivative. The mentioned derivative has been introduced by Atangana, Baleanu, and Caputo abbreviated as$(\mathcal{ABC})$(ABC). Upon utilizing fixed point theory, we first prove the existence of at least one solution for the considered model and its uniqueness. Also, some results about stability of Ulam–Hyers type are also established. By applying a numerical technique called fractional Adams–Bashforth (AB) method, we develop a scheme for the approximate solutions to the considered model. Using some real available data, we perform the concerned numerical simulation corresponding to different values of fractional order.
This manuscript is concerned to the existence and stability of HIV/AIDS infection model with frac... more This manuscript is concerned to the existence and stability of HIV/AIDS infection model with fractional order derivative. The corresponding derivative is taken in Caputo-Fabrizio sense, which is a new approach for such type of biological models. With the help of Sumudu transform, some new results are handled. Further for the corresponding results, existence theory and uniqueness for the equilibrium solution are provided via using nonlinear functional analysis and fixed point theory due to Banach.
International Journal of Applied and Computational Mathematics, 2020
In this research work, we considered the numerical solution of an essential fractional dynamical ... more In this research work, we considered the numerical solution of an essential fractional dynamical model of HIV infection of CD4 + T cells, by using a relatively new technique, Chebyshev Wavelet Method. The numerical simulations are performed for integer-order as well as for fractional-order of HIV models. In this research work, the fractional derivatives are defined by Caputo operator. The proposed method is implemented to solve both fractional and integer-order of HIV models. The result obtained by the suggested method are compare with Müntz-Legendre Polynomials, Laplace Transform with Padé Approximation, Developed Bessel Collocation Method, Laplace Decomposition Method and Runge-Kutta fourth-order (RK4) method. The comparison has verified the greatest accuracy then other existing method.
In this paper, optimal homotopy asymptotic method has been extended to seek out the exact solutio... more In this paper, optimal homotopy asymptotic method has been extended to seek out the exact solution of fractional generalized biological population models. The time fractional derivatives are described in the Caputo sense. It optimal homotopy asymptotic method is a new approach for fractional models. The proposed approach presents a procedure by that we have transferred the model to a series of simpler problems which are solvable by hand work applying the Riemann-Liouville fractional integral operator and obtained exact solution of fractional the generalized biological population by adding the solutions of first three simple problems of the series of simpler problems. The new approach provides exact solution in the way of smoothly convergent series.
Uploads
Papers by Samia Bushnaq