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Discrete & Continuous Dynamical Systems - S

June 2019 , Volume 12 , Issue 3

Issue on recent development in numerical and analytical methods

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Preface: New trends on numerical analysis and analytical methods with their applications to real world problems
Abdon Atangana, Zakia Hammouch, Kolade M. Owolabi and Gisele Mephou
2019, 12(3): ⅰ-ⅰ doi: 10.3934/dcdss.201903i +[Abstract](71) +[HTML](37) +[PDF](74.28KB)
Abstract:
Generalised class of Time Fractional Black Scholes equation and numerical analysis
Rodrigue Gnitchogna Batogna and Abdon Atangana
2019, 12(3): 435-445 doi: 10.3934/dcdss.2019028 +[Abstract](71) +[HTML](47) +[PDF](377.35KB)
Abstract:

It is well known now, that a Time Fractional Black Scholes Equation (TFBSE) with a time derivative of real order \begin{document}$ \alpha $\end{document} can be obtained to describe the price of an option, when for example the change in the underlying asset is assumed to follow a fractal transmission system. Fractional derivatives as they are called were introduced in option pricing in a bid to take advantage of their memory properties to capture both major jumps over small time periods and long range dependencies in markets. Recently new derivatives of Fractional Calculus with non local and/or non singular Kernel, have been introduced and have had substantial changes in modelling of some diffusion processes. Based on consistency and heuristic arguments, this work generalises previously obtained Time Fractional Black Scholes Equations to a new class of Time Fractional Black Scholes Equations. A numerical scheme solution is also derived. The stability of the numerical scheme is discussed, graphical simulations are produced to price a double barriers knock out call option.

New exact solutions for some fractional order differential equations via improved sub-equation method
Berat Karaagac
2019, 12(3): 447-454 doi: 10.3934/dcdss.2019029 +[Abstract](83) +[HTML](33) +[PDF](846.56KB)
Abstract:

In this paper, improved sub-equation method is proposed to obtain new exact analytical solutions for some nonlinear fractional differential equations by means of modified Riemann Liouville derivative. The method is applied to time-fractional biological population model and space-time fractional Fisher equation successfully. Finally, simulations of new exact analytical solutions are presented graphically.

Modeling the transmission dynamics of avian influenza with saturation and psychological effect
Muhammad Altaf Khan, Muhammad Farhan, Saeed Islam and Ebenezer Bonyah
2019, 12(3): 455-474 doi: 10.3934/dcdss.2019030 +[Abstract](149) +[HTML](105) +[PDF](1679.39KB)
Abstract:

The present paper describes the mathematical analysis of an avian influenza model with saturation and psychological effect. The virus of avian influenza is not only a risk for birds but the population of human is also not safe from this. We proposed two models, one for birds and the other one for human. We consider saturated incidence rate and psychological effect in the model. The stability results for each model that is birds and human is investigated. The local and global dynamics for the disease free case of each model is proven when the basic reproduction number \begin{document}$ \mathcal{R}_{0b}<1$\end{document} and \begin{document}$ \mathcal{R}_0<1$\end{document}. Further, the local and global stability of each model is investigated in the case when \begin{document}$ \mathcal{R}_{0b}>1$\end{document} and \begin{document}$ \mathcal{R}_0>1$\end{document}. The mathematical results show that the considered saturation effect in population of birds and psychological effect in population of human does not effect the stability of equilibria, if the disease is prevalent then it can affect the number of infected humans. Numerical results are carried out in order to validate the theoretical results. Some numerical results for the proposed parameters are presented which can reduce the number of infective in the population of humans.

Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative
Ilknur Koca
2019, 12(3): 475-486 doi: 10.3934/dcdss.2019031 +[Abstract](67) +[HTML](45) +[PDF](322.4KB)
Abstract:

A nonlinear system of two fractional nonlinear differential equations with Atangana-Baleanu derivative is considered in this work. General conditions under which a system solution exists and unique are presented using the fixed-point theorem method. The well-established numerical scheme is used to solve the system of equations. A numerical analysis is presented to secure the stability and convergence of the used numerical scheme.

The first integral method for two fractional non-linear biological models
Olusola Kolebaje, Ebenezer Bonyah and Lateef Mustapha
2019, 12(3): 487-502 doi: 10.3934/dcdss.2019032 +[Abstract](71) +[HTML](100) +[PDF](2681.95KB)
Abstract:

Travelling wave solutions of the space and time fractional models for non-linear blood flow in large vessels and Deoxyribonucleic acid (DNA) molecule dynamics defined in the sense of Jumarie's modified Riemann-Liouville derivative via the first integral method are presented in this study. A fractional complex transformation was applied to turn the fractional biological models into an equivalent integer order ordinary differential equation. The validity of the solutions to the fractional biological models obtained with first integral method was achieved by putting them back into the models. We observed that introducing fractional order to the biological models changes the nature of the solution.

On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation
Gökçe Dİlek Küçük, Gabil Yagub and Ercan Çelİk
2019, 12(3): 503-512 doi: 10.3934/dcdss.2019033 +[Abstract](63) +[HTML](26) +[PDF](364.62KB)
Abstract:

In this paper, an optimal control problem for Schrödinger equation with complex coefficient which contains gradient is examined. A theorem is given that states the existence and uniqueness of the solution of the initial-boundary value problem for Schrödinger equation. Then for the solution of the optimal control problem, two different cases are investigated. Firstly, it is shown that the optimal control problem has a unique solution for \begin{document}$α >0$\end{document} on a dense subset $G$ on the space $H$ which contains the measurable square integrable functions on \begin{document}$\left(0,l\right)$\end{document} and secondly the optimal control problem has at least one solution for any \begin{document}$α ≥ 0$\end{document} on the space $H$.

Weak Galerkin mixed finite element methods for parabolic equations with memory
Xiaomeng Li, Qiang Xu and Ailing Zhu
2019, 12(3): 513-531 doi: 10.3934/dcdss.2019034 +[Abstract](47) +[HTML](40) +[PDF](442.02KB)
Abstract:

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both \begin{document}$ |\|·|\| $\end{document} and \begin{document}$ L^2 $\end{document} norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces \begin{document}$ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $\end{document} for \begin{document}$ k = 0 $\end{document} and the numerical convergence rates confirm the theoretical results.

A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation
Ömer Oruç, Alaattin Esen and Fatih Bulut
2019, 12(3): 533-542 doi: 10.3934/dcdss.2019035 +[Abstract](76) +[HTML](37) +[PDF](414.65KB)
Abstract:

In this paper, we developed a unified method to solve time fractional Burgers' equation using the Chebyshev wavelet and L1 discretization formula. First we give the preliminary information about Chebyshev wavelet method, then we describe time discretization of the problems under consideration and then we apply Chebyshev wavelets for space discretization. The performance of the method is shown by three test problems and obtained results compared with other results available in literature.

Numerical analysis and pattern formation process for space-fractional superdiffusive systems
Kolade M. Owolabi
2019, 12(3): 543-566 doi: 10.3934/dcdss.2019036 +[Abstract](85) +[HTML](31) +[PDF](1904.83KB)
Abstract:

In this paper, we consider the numerical solution of fractional-in-space reaction-diffusion equation, which is obtained from the classical reaction-diffusion equation by replacing the second-order spatial derivative with a fractional derivative of order \begin{document}$ α∈(1, 2] $\end{document}. We adopt a class of second-order approximations, based on the weighted and shifted Grünwald difference operators in Riemann-Liouville sense to numerically simulate two multicomponent systems with fractional-order in higher dimensions. The efficiency and accuracy of the numerical schemes are justified by reporting the norm infinity and norm relative errors as well as their convergence. The complexity of the dynamics in the equation is theoretically discussed by conducting its local and global stability analysis and Numerical experiments are performed to back-up the theoretical claims.

High-order solvers for space-fractional differential equations with Riesz derivative
Kolade M. Owolabi and Abdon Atangana
2019, 12(3): 567-590 doi: 10.3934/dcdss.2019037 +[Abstract](87) +[HTML](26) +[PDF](2009.11KB)
Abstract:

This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order \begin{document}$ α $\end{document} in \begin{document}$ (0, 2] $\end{document}. The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics
Kolade M. Owolabi, Kailash C. Patidar and Albert Shikongo
2019, 12(3): 591-613 doi: 10.3934/dcdss.2019038 +[Abstract](68) +[HTML](36) +[PDF](536.0KB)
Abstract:

In this paper, we extend a system of coupled first order non-linear system of delay differential equations (DDEs) arising in modeling of stoichiometry of tumour dynamics, to a system of diffusion-reaction system of partial delay differential equations (PDDEs). Since tumor cells is further modified by blood supply through the vascularization process, thus we determine the local uniform steady states of the homogeneous tumour growth model with respect to the vascularization process. We show that the steady states are globally stable, determine the existence of Hopf bifurcation of the homogeneous tumour growth model with respect to the vascularization process. We derive, analyse and implement a fitted operator finite difference method (FOFDM) to solve the extended model. This FOFDM is analyzed for convergence and it is seen that it has second-order accuracy. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.

The discrete homotopy perturbation Sumudu transform method for solving partial difference equations
Figen Özpinar and Fethi Bin Muhammad Belgacem
2019, 12(3): 615-624 doi: 10.3934/dcdss.2019039 +[Abstract](49) +[HTML](26) +[PDF](6929.15KB)
Abstract:

In this paper, we introduce a combined form of the discrete Sumudu transform method with the discrete homotopy perturbation method to solve linear and nonlinear partial difference equations. This method is called the discrete homotopy perturbation Sumudu transform method(DHPSTM). The results reveal that the introduced method is very efficient, simple and can be applied to other linear and nonlinear difference equations. The nonlinear terms can be easily handled by use of He's polynomials.

Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models
Edson Pindza, Francis Youbi, Eben Maré and Matt Davison
2019, 12(3): 625-643 doi: 10.3934/dcdss.2019040 +[Abstract](113) +[HTML](24) +[PDF](636.33KB)
Abstract:

A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Lévy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.

Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel
Muhammad Bilal Riaz, Naseer Ahmad Asif, Abdon Atangana and Muhammad Imran Asjad
2019, 12(3): 645-664 doi: 10.3934/dcdss.2019041 +[Abstract](59) +[HTML](25) +[PDF](821.22KB)
Abstract:

Couette flows of an incompressible viscous fluid with non-integer order derivative without singular kernel produced by the motion of a flat plate are analyzed under the slip condition at boundaries. An analytical transform approach is used to obtain the exact expressions for velocity and shear stress. Three particular cases from the general results with and without slip at the wall are obtained. These solutions, which are organized in simple forms in terms of exponential and trigonometric functions, can be conveniently engaged to obtain known solutions from the literature. The control of the new non-integer order derivative on the velocity of the fluid moreover a comparative study with an older model, is analyzed for some flows with practical applications. The non-integer order derivative with non-singular kernel is more appropriate for handling mathematical calculations of obtained solutions.

Comparative study of a cubic autocatalytic reaction via different analysis methods
Khaled Mohammed Saad and Eman Hussain Faissal AL-Sharif
2019, 12(3): 665-684 doi: 10.3934/dcdss.2019042 +[Abstract](119) +[HTML](32) +[PDF](2057.25KB)
Abstract:

In this paper we discuss an approximate solutions of the space-time fractional cubic autocatalytic chemical system (STFCACS) equations. The main objective is to find and compare approximate solutions of these equations found using Optimal q-Homotopy Analysis Method (Oq-HAM), Homotopy Analysis Transform Method (HATM), Varitional Iteration Method (VIM) and Adomian Decomposition Method (ADM).

Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel
Yin Yang and Yunqing Huang
2019, 12(3): 685-702 doi: 10.3934/dcdss.2019043 +[Abstract](56) +[HTML](25) +[PDF](489.43KB)
Abstract:

We consider spectral and pseudo-spectral Jacobi-Galerkin methods and corresponding iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. The Gauss-Jacobi quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations, which show that the errors of the approximate solution decay exponentially in \begin{document}$L^∞$\end{document}-norm and weighted \begin{document}$L^2$\end{document}-norm. The numerical examples are given to illustrate the theoretical results.

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