In this paper, the global stability analysis of the pertussis model with maternally derived immun... more In this paper, the global stability analysis of the pertussis model with maternally derived immunity is studied. The model is qualitatively analysed to investigate the asymptotic behavior of the model with respect to the equilibria. Using the Lyapunov function method, the pertussis-free equilibrium is determined to be globally asymptotically stable when the associated basic reproduction number is less than unity. Furthermore, the study is also extended to prove the existence of a globally asymptotically stable pertussis endemic equilibrium using a suitable nonlinear Lyapunov function.
In this paper, we examined a model of cell invasion focusing on the wavefront of the neural crest... more In this paper, we examined a model of cell invasion focusing on the wavefront of the neural crest (NC) cells in the case of Hirschsprung's disease (HSCR). Hirschsprung's disease (HSCR) is a congenital defect of intestinal ganglion cells and causes patients to have disorders in peristalsis. This simulation model was performed using the fractional differential equations (FDEs) based upon two basic cell functions. Here, we simulated the mathematical model in a one-dimensional setting, based on the fractional trapezoidal numerical scheme and the results showed an interesting outcome for the mobility of the cellular processes under crowded environments.
In this paper, two high order compact finite difference schemes are formulated for solving the on... more In this paper, two high order compact finite difference schemes are formulated for solving the one dimensional anomalous subdiffusion equation. The Grünwald-Letnikov formula is used to discretize the temporal fractional derivative. The truncation error and stability of the two methods are discussed. The feasibility of the compact schemes is investigated by application to a model problem.
There are a number of physical situations that can be modeled by fractional partial differential ... more There are a number of physical situations that can be modeled by fractional partial differential equations. In this paper, we discuss a numerical scheme based on Keller's box method for one dimensional time fractional diffusion equation with boundary values which are functions. The fractional derivative term is replaced by the Grünwald-Letnikov formula. The stability is analyzed by means of the Von Neumann method. An example is presented to show the feasibility and the accuracy of this method and a comparison between the approximate solution using this method and analytical solution is made. The results indicated that this scheme is unconditionally stable and is a feasible technique.
In this article, we present a variant approach of the generalized and improved (G'/G)-expansion m... more In this article, we present a variant approach of the generalized and improved (G'/G)-expansion method and construct some new exact traveling wave solutions with free parameters of the nonlinear evolution equations, via the Painleve integrable Burgers equation, the Boiti-Leon-Pempinelle equation and the Pochhammer-Chree equations. When the free parameters receive special values, solitons are originated from the travelling wave. In the new approach, G() satisfies the Jacobi elliptic equation in place of the second order linear equation. It is shown that the suggested algorithm is quite efficient and is practically well suited to solve these problems.
Journal of Interpolation and Approximation in Scientific Computing, 2017
A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimension... more A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grünwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible.
In this paper three implicit finite difference methods are developed to solve one dimensional tim... more In this paper three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation with boundary values which are functions. The temporal fractional derivative is approximated by using Grünwald-Letnikov formula of order α∈(0,1) while the first and second order spatial derivatives are discretized by applying the compact and high order finite difference approximation. The local truncation error was analyzed and it is shown that all threeschemes have fourth order accuracy in space and first order in time. The stability of all the schemes was investigated using the von Neumann method and mathematical induction and it was found that all the implicit schemes are unconditionally.
Despite recent advances in the mathematical modeling of biological processes and real-world situa... more Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential equations seems to be a quite promising and appropriate tool to model such complex and nonlinear phenomena. Fractional differential equations have recently gained a significant deal of attention and demonstrated their relevance in modeling real phenomena rather than their counterpart, classical (integer) derivative differential equations. We report in this paper a mathematical approach susceptible to answering some relevant questions regarding the side effects of ionizing radiation (IR) on DNA with a particular focus on double-strand breaks (DSBs), leading to the destruction of the cell population. A theoretical elucidation of the population memory was carried out wit...
Determining the non-linear traveling or soliton wave solutions for variable-order fractional evol... more Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.
This paper addresses the numerical study of variable-order fractional differential equation based... more This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.
The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractiona... more The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractional diffusion equation by means of cubic B-spline functions. The Caputo–Fabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effectively. The proposed numerical technique relies on finite difference approach and cubic B-spline functions for discretization along temporal and spatial grids, respectively. To ensure that the error does not amplify during computational process, stability analysis is performed. The described algorithm is second-order convergent along time and space directions. The computational competence of the scheme is tested through some numerical examples. The results reveal that the current scheme is reasonably efficient and reliable to be used for solving the subject problem.
Fractional differential equations describe nature adequately because of the symmetry properties t... more Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate.
A susceptible-infected-recovered compartmental model incorporating maternally derived immunity co... more A susceptible-infected-recovered compartmental model incorporating maternally derived immunity compartment is analyzed in this study. The stability of the pertussis-disease-free and endemic equilibrium is studied. The basic reproduction number is obtained and its behavior analyzed by varying parameters. Numerical simulations indicated that when the waning parameter is increased, the frequency at which the population attains stability varies. However, the infected population does not go extinct even at equilibrium.
In this article, the improved ( ) / GG ' -expansion method has been implemented to generate t... more In this article, the improved ( ) / GG ' -expansion method has been implemented to generate travelling wave solutions, where ( ) G ξ satisfies the second order linear ordinary differential equation. To show the advantages of the method, the (3+1)-dimensional Kadomstev-Petviashvili (KP) equation has been investigated. Higher- dimensional nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation. Also, in order to understand the behaviour of solutions, the graphical representations of some obtained solutions have been presented.
In this article, the Exp -function method is applied to construct traveling wave solutions of the... more In this article, the Exp -function method is applied to construct traveling wave solutions of the fifth-order Sawada-Kotera equation. This method is one of the powerful methods that appear in recent time for establishing exact traveling wave solutions of nonlinear partial differential equations. The solution procedure of this method is implemented by symbolic software, such as, Maple. We obtain some new exact solutions including solitary and periodic wave solutions. It is shown that the Exp -function method is straightforward and effective mathematical tool for solving nonlinear evolution equations in mathematical physics and engineering sciences. In addition, some of solutions are described in the figures with the aid of commercial software Maple.
In this paper, a mathematical model of dengue incorporating two sub-models that: describes the li... more In this paper, a mathematical model of dengue incorporating two sub-models that: describes the linked dynamics between predator-prey of mosquitoes at the larval stage, and describes the dengue spread between humans and adult mosquitoes, is formulated to simulate the dynamics of dengue spread. The effect of predator-prey dynamics in controlling the dengue disease at the larval stage of mosquito populations is investigated. Stability analysis of the equilibrium points are carried out. Numerical simulations results indicate that the use of predator-prey dynamics of mosquitoes at the larval stage as biological control agents for controlling the larval stage of dengue mosquito assists in combating dengue virus contagion.
The improved (G’/G)-expansion method is a powerful mathematical tool for solving nonlinear evolut... more The improved (G’/G)-expansion method is a powerful mathematical tool for solving nonlinear evolution equations whicharise in mathematical physics, engineering sciences and other technical arena. In this article, we construct some new exact travelingwave solutions for the modified Benjamin-Bona-Mahony equation by applying the improved (G’/G)-expansion method. In themethod, the general solution of the second order linear ordinary differential equation with constant coefficients is used for studyingnonlinear partial differential equations. The solution procedure of this method is executed by algebraic software, such as, Maple. Theobtained solutions including solitary and periodic wave solutions are presented in terms of the hyperbolic function, the trigonometricfunction and the rational forms. It is noteworthy to reveal that some of our solutions are in good agreement with the published resultsfor special cases which certifies our other solutions. Furthermore, the graphical presentatio...
The behavior of quintic nonlinear dispersion coefficient of creeping soliton in a spatial domain ... more The behavior of quintic nonlinear dispersion coefficient of creeping soliton in a spatial domain with hyperbolicity analysis of Hartman-Grobman Theorem by using variational approach is studied. Complex Ginzburg-Landau equation (CGLE) is used in the analysis as we relate the creeping soliton with Hartman- Grobman Theorem. We evaluated our work based on perturbed Jacobian matrix from system of three supercritical ordinary differential Euler-Lagrange equations, in which the eigenvalues of the stability matrix touch the imaginary axis. As a consequence in unfolding the bifurcation of creeping solitons, the equilibrium structure ultimately chaotic at the variation of the coefficient µ away from the critical value, µc . This leads to hyperbolicity loss of Hartman-Grobman Theorem in the dissipative system driven out the oscillatory instability of µ exceeded the criticality parameter corresponding to the Hopf bifurcations as the system is highly complex. This overall approach restrict to nu...
In this paper, the global stability analysis of the pertussis model with maternally derived immun... more In this paper, the global stability analysis of the pertussis model with maternally derived immunity is studied. The model is qualitatively analysed to investigate the asymptotic behavior of the model with respect to the equilibria. Using the Lyapunov function method, the pertussis-free equilibrium is determined to be globally asymptotically stable when the associated basic reproduction number is less than unity. Furthermore, the study is also extended to prove the existence of a globally asymptotically stable pertussis endemic equilibrium using a suitable nonlinear Lyapunov function.
In this paper, we examined a model of cell invasion focusing on the wavefront of the neural crest... more In this paper, we examined a model of cell invasion focusing on the wavefront of the neural crest (NC) cells in the case of Hirschsprung's disease (HSCR). Hirschsprung's disease (HSCR) is a congenital defect of intestinal ganglion cells and causes patients to have disorders in peristalsis. This simulation model was performed using the fractional differential equations (FDEs) based upon two basic cell functions. Here, we simulated the mathematical model in a one-dimensional setting, based on the fractional trapezoidal numerical scheme and the results showed an interesting outcome for the mobility of the cellular processes under crowded environments.
In this paper, two high order compact finite difference schemes are formulated for solving the on... more In this paper, two high order compact finite difference schemes are formulated for solving the one dimensional anomalous subdiffusion equation. The Grünwald-Letnikov formula is used to discretize the temporal fractional derivative. The truncation error and stability of the two methods are discussed. The feasibility of the compact schemes is investigated by application to a model problem.
There are a number of physical situations that can be modeled by fractional partial differential ... more There are a number of physical situations that can be modeled by fractional partial differential equations. In this paper, we discuss a numerical scheme based on Keller's box method for one dimensional time fractional diffusion equation with boundary values which are functions. The fractional derivative term is replaced by the Grünwald-Letnikov formula. The stability is analyzed by means of the Von Neumann method. An example is presented to show the feasibility and the accuracy of this method and a comparison between the approximate solution using this method and analytical solution is made. The results indicated that this scheme is unconditionally stable and is a feasible technique.
In this article, we present a variant approach of the generalized and improved (G'/G)-expansion m... more In this article, we present a variant approach of the generalized and improved (G'/G)-expansion method and construct some new exact traveling wave solutions with free parameters of the nonlinear evolution equations, via the Painleve integrable Burgers equation, the Boiti-Leon-Pempinelle equation and the Pochhammer-Chree equations. When the free parameters receive special values, solitons are originated from the travelling wave. In the new approach, G() satisfies the Jacobi elliptic equation in place of the second order linear equation. It is shown that the suggested algorithm is quite efficient and is practically well suited to solve these problems.
Journal of Interpolation and Approximation in Scientific Computing, 2017
A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimension... more A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grünwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible.
In this paper three implicit finite difference methods are developed to solve one dimensional tim... more In this paper three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation with boundary values which are functions. The temporal fractional derivative is approximated by using Grünwald-Letnikov formula of order α∈(0,1) while the first and second order spatial derivatives are discretized by applying the compact and high order finite difference approximation. The local truncation error was analyzed and it is shown that all threeschemes have fourth order accuracy in space and first order in time. The stability of all the schemes was investigated using the von Neumann method and mathematical induction and it was found that all the implicit schemes are unconditionally.
Despite recent advances in the mathematical modeling of biological processes and real-world situa... more Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential equations seems to be a quite promising and appropriate tool to model such complex and nonlinear phenomena. Fractional differential equations have recently gained a significant deal of attention and demonstrated their relevance in modeling real phenomena rather than their counterpart, classical (integer) derivative differential equations. We report in this paper a mathematical approach susceptible to answering some relevant questions regarding the side effects of ionizing radiation (IR) on DNA with a particular focus on double-strand breaks (DSBs), leading to the destruction of the cell population. A theoretical elucidation of the population memory was carried out wit...
Determining the non-linear traveling or soliton wave solutions for variable-order fractional evol... more Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.
This paper addresses the numerical study of variable-order fractional differential equation based... more This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.
The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractiona... more The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractional diffusion equation by means of cubic B-spline functions. The Caputo–Fabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effectively. The proposed numerical technique relies on finite difference approach and cubic B-spline functions for discretization along temporal and spatial grids, respectively. To ensure that the error does not amplify during computational process, stability analysis is performed. The described algorithm is second-order convergent along time and space directions. The computational competence of the scheme is tested through some numerical examples. The results reveal that the current scheme is reasonably efficient and reliable to be used for solving the subject problem.
Fractional differential equations describe nature adequately because of the symmetry properties t... more Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate.
A susceptible-infected-recovered compartmental model incorporating maternally derived immunity co... more A susceptible-infected-recovered compartmental model incorporating maternally derived immunity compartment is analyzed in this study. The stability of the pertussis-disease-free and endemic equilibrium is studied. The basic reproduction number is obtained and its behavior analyzed by varying parameters. Numerical simulations indicated that when the waning parameter is increased, the frequency at which the population attains stability varies. However, the infected population does not go extinct even at equilibrium.
In this article, the improved ( ) / GG ' -expansion method has been implemented to generate t... more In this article, the improved ( ) / GG ' -expansion method has been implemented to generate travelling wave solutions, where ( ) G ξ satisfies the second order linear ordinary differential equation. To show the advantages of the method, the (3+1)-dimensional Kadomstev-Petviashvili (KP) equation has been investigated. Higher- dimensional nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation. Also, in order to understand the behaviour of solutions, the graphical representations of some obtained solutions have been presented.
In this article, the Exp -function method is applied to construct traveling wave solutions of the... more In this article, the Exp -function method is applied to construct traveling wave solutions of the fifth-order Sawada-Kotera equation. This method is one of the powerful methods that appear in recent time for establishing exact traveling wave solutions of nonlinear partial differential equations. The solution procedure of this method is implemented by symbolic software, such as, Maple. We obtain some new exact solutions including solitary and periodic wave solutions. It is shown that the Exp -function method is straightforward and effective mathematical tool for solving nonlinear evolution equations in mathematical physics and engineering sciences. In addition, some of solutions are described in the figures with the aid of commercial software Maple.
In this paper, a mathematical model of dengue incorporating two sub-models that: describes the li... more In this paper, a mathematical model of dengue incorporating two sub-models that: describes the linked dynamics between predator-prey of mosquitoes at the larval stage, and describes the dengue spread between humans and adult mosquitoes, is formulated to simulate the dynamics of dengue spread. The effect of predator-prey dynamics in controlling the dengue disease at the larval stage of mosquito populations is investigated. Stability analysis of the equilibrium points are carried out. Numerical simulations results indicate that the use of predator-prey dynamics of mosquitoes at the larval stage as biological control agents for controlling the larval stage of dengue mosquito assists in combating dengue virus contagion.
The improved (G’/G)-expansion method is a powerful mathematical tool for solving nonlinear evolut... more The improved (G’/G)-expansion method is a powerful mathematical tool for solving nonlinear evolution equations whicharise in mathematical physics, engineering sciences and other technical arena. In this article, we construct some new exact travelingwave solutions for the modified Benjamin-Bona-Mahony equation by applying the improved (G’/G)-expansion method. In themethod, the general solution of the second order linear ordinary differential equation with constant coefficients is used for studyingnonlinear partial differential equations. The solution procedure of this method is executed by algebraic software, such as, Maple. Theobtained solutions including solitary and periodic wave solutions are presented in terms of the hyperbolic function, the trigonometricfunction and the rational forms. It is noteworthy to reveal that some of our solutions are in good agreement with the published resultsfor special cases which certifies our other solutions. Furthermore, the graphical presentatio...
The behavior of quintic nonlinear dispersion coefficient of creeping soliton in a spatial domain ... more The behavior of quintic nonlinear dispersion coefficient of creeping soliton in a spatial domain with hyperbolicity analysis of Hartman-Grobman Theorem by using variational approach is studied. Complex Ginzburg-Landau equation (CGLE) is used in the analysis as we relate the creeping soliton with Hartman- Grobman Theorem. We evaluated our work based on perturbed Jacobian matrix from system of three supercritical ordinary differential Euler-Lagrange equations, in which the eigenvalues of the stability matrix touch the imaginary axis. As a consequence in unfolding the bifurcation of creeping solitons, the equilibrium structure ultimately chaotic at the variation of the coefficient µ away from the critical value, µc . This leads to hyperbolicity loss of Hartman-Grobman Theorem in the dissipative system driven out the oscillatory instability of µ exceeded the criticality parameter corresponding to the Hopf bifurcations as the system is highly complex. This overall approach restrict to nu...
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