# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

March 2020 , Volume 13 , Issue 3

Issue on new trends of numerical and analytical methods

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2020, 13(3): ⅰ-ⅱ doi: 10.3934/dcdss.20203i +[Abstract](1247) +[HTML](99) +[PDF](75.82KB)
Abstract:
2020, 13(3): 351-375 doi: 10.3934/dcdss.2020020 +[Abstract](3305) +[HTML](613) +[PDF](478.68KB)
Abstract:

Recently, U. N. Katugampola presented some generalized fractional integrals and derivatives by iterating a \begin{document}$t^{\rho-1}-$\end{document}weighted integral, \begin{document}$\rho>0$\end{document}. The case \begin{document}$\rho = 1$\end{document} produces Riemann and Caputo fractional derivatives and the limiting case \begin{document}$\rho\rightarrow 0^+$\end{document} results in Hadamard type fractional operators. In this article, we discuss the differences between a new class of nonlocal generalized fractional derivatives generated by iterating left and right type conformable integrals weighted by \begin{document}$(t-a)^{\rho-1}$\end{document} and \begin{document}$(b-t)^{\rho-1}$\end{document} and the ones introduced by Katugampola. In fact, we will present very different integration by parts formulas by presenting new mixed left and right generalized fractional operators with boundary points dependent kernels. The properties of this new class of mixed fractional operators are analyzed in newly defined function spaces as well.

2020, 13(3): 377-387 doi: 10.3934/dcdss.2020021 +[Abstract](2737) +[HTML](848) +[PDF](336.34KB)
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The novelty of this research is to utilize the modern approach of Atangana-Baleanu fractional derivative to electrically conducting viscous fluid embedded in porous medium. The mathematical modeling of the governing partial differential equations is characterized via non-singular and non-local kernel. The set of governing fractional partial differential equations is solved by employing Laplace transform technique. The analytic solutions are investigated for the velocity field corresponding with shear stress and expressed in term of special function namely Fox-H function, moreover a comparative study with an ordinary and Atangana-Baleanu fractional models is analyzed for viscous flow in presence and absence of magnetic field and porous medium. The Atangana-Baleanu fractional derivative is observed more reliable and appropriate for handling mathematical calculations of obtained solutions. Finally, graphical illustration is depicted via embedded rheological parameters and comparison of models plotted for smaller and larger time on the fluid flow.

2020, 13(3): 389-406 doi: 10.3934/dcdss.2020022 +[Abstract](2313) +[HTML](649) +[PDF](501.68KB)
Abstract:

The complexity and non-linearity of flow phenomena are explained by numerous criteria, including the interactions of the large number of vehicles occupying the road, which influence the road density. This density under certain conditions, leads to traffic congestion which has dangerous effects on the environment such as; resources consumption; noise and the effect caused by greenhouse gas emissions of the \begin{document}$CO_{2}$\end{document} and other pollutants. In this paper we consider working in an uniform, homogeneous road where the traffic is described by the Lighthill Whitham-Richard (LWR) model resolved using a cubic B-spline collocation scheme in space and an implicit Runge Kutta scheme in time. We also shed light on the relation between vehicle occupancy and vehicle emissions.

2020, 13(3): 407-428 doi: 10.3934/dcdss.2020023 +[Abstract](3907) +[HTML](1239) +[PDF](767.59KB)
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This paper deals with a new formulation of time fractional optimal control problems governed by Caputo-Fabrizio (CF) fractional derivative. The optimality system for this problem is derived, which contains the forward and backward fractional differential equations in the sense of CF. These equations are then expressed in terms of Volterra integrals and also solved by a new numerical scheme based on approximating the Volterra integrals. The linear rate of convergence for this method is also justified theoretically. We present three illustrative examples to show the performance of this method. These examples also test the contribution of using CF derivative for dynamical constraints and we observe the efficiency of this new approach compared to the classical version of fractional operators.

2020, 13(3): 429-442 doi: 10.3934/dcdss.2020024 +[Abstract](2134) +[HTML](603) +[PDF](350.31KB)
Abstract:

A new concept of dynamical system of predator-prey model is presented in this paper. The model takes into account the memory of interaction between the prey and predator due to the inclusion of fractional differentiation. In addition, the model takes into account the inherent disposition of a prey or predator toward hunting or defending in time. Analysis of existence and uniqueness of the solutions is presented. A numerical method is used to generate some simulations as the fractional orders change from one to zero. A new traveling waves patterns are obtained.

2020, 13(3): 443-466 doi: 10.3934/dcdss.2020025 +[Abstract](1924) +[HTML](739) +[PDF](367.52KB)
Abstract:

The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established.

2020, 13(3): 467-484 doi: 10.3934/dcdss.2020026 +[Abstract](2726) +[HTML](572) +[PDF](2782.48KB)
Abstract:

The concept of differentiation with power law reset has not been investigated much in the literature, as some researchers believe the concept was wrongly introduced, it is unable to describe fractal sharps. It is important to note that, this concept of differentiation is not to describe or display fractal sharps but to describe a flow within a medium with self-similar properties. For instance, the description of flow within a non-conventional media which does not obey the classical Fick's laws of diffusion, Darcy's law and Fourier's law cannot be handle accurately with conventional mechanical law of rate of change. In this paper, we pointed out the use of the non-conventional differential operator with fractal dimension and it possible applicability in several field of sciences, technology and engineering dealing with non-conventional flow. Due to the wider applicability of this concept and the complexities of solving analytically those partial differential equations generated from this operator, we introduced in this paper a new numerical scheme that will be able to handle this class of differential equations. We presented in general the conditions of stability and convergence of the numerical scheme. We applied to some well-known diffusion and subsurface flow models and the stability analysis and numerical simulations for each cases are presented.

2020, 13(3): 485-501 doi: 10.3934/dcdss.2020027 +[Abstract](2933) +[HTML](738) +[PDF](414.85KB)
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In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana-Baleanu derivatives. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's derivatives with multi parameters are investigated. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP.

2020, 13(3): 503-518 doi: 10.3934/dcdss.2020028 +[Abstract](2564) +[HTML](673) +[PDF](554.52KB)
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In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations via perturbation iteration technique and newly developed optimal perturbation iteration method. Some specific examples are given and obtained solutions are compared with other methods and analytical results to confirm the good accuracy of the proposed methods.We also discuss the convergence of the optimal perturbation iteration method for partial differential equations. The results reveal that perturbation iteration techniques, unlike many other techniques in literature, converge rapidly to exact solutions of the given problems at lower order of approximations.

2020, 13(3): 519-537 doi: 10.3934/dcdss.2020029 +[Abstract](2407) +[HTML](660) +[PDF](607.29KB)
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In this paper, Lymphatic filariasis-schistosomiasis coinfected model is studied within the context of fractional derivative order based on Mittag-Leffler function of ABC in the Caputo sense. The existence and uniqueness of system model solution is derived by employing a well- known Banach fixed point theorem. The numerical solution based on the Mittag-Leffler function suggests that the dynamics of the coinfected model is well explored using fractional derivative order because of non-singularity.

2020, 13(3): 539-560 doi: 10.3934/dcdss.2020030 +[Abstract](3018) +[HTML](1061) +[PDF](351.76KB)
Abstract:

The aim of the present paper is to establish certain new image formulae of family of some extended generalized Gauss hypergeometric functions by applying the operators of fractional derivative involving \begin{document}${}_2F_1(.)$\end{document} due to Saigo. Furthermore, by employing some integral transforms on the resulting formulas, we obtained some more image formulas and also develop a new and further generalized form of the fractional kinetic equation involving the family of some extended generalized Gauss hypergeometric functions and the manifold generality of the family of functions is discussed in terms of the solution of the fractional kinetic equation. The results obtained here are quite general in nature.

2020, 13(3): 561-574 doi: 10.3934/dcdss.2020031 +[Abstract](2988) +[HTML](774) +[PDF](860.14KB)
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In this work we present a numerical method based on the Adams-Bashforth-Moulton scheme to solve numerically fractional delay differential equations. We focus in the fractional derivative with Mittag-Leffler kernel of type Liouville-Caputo with variable-order and the Liouville-Caputo fractional derivative with variable-order. Numerical examples are presented to show the applicability and efficiency of this novel method.

2020, 13(3): 575-607 doi: 10.3934/dcdss.2020032 +[Abstract](2344) +[HTML](795) +[PDF](608.23KB)
Abstract:

We have constructed the basic dielectric relaxations evolution expression for relaxing current as convolution operation of the chosen memory kernel and rate of change of applied voltage. We have studied types of memory kernels singular, non-singular and combination of singular and non-singular (mixed) decaying functions. With these, we form constitutive equations for relaxation dynamics of dielectrics; i.e. capacitor. We observe that though mathematically we can use non-singular kernels yet this does not give presently much useful practical or physically realizable results and interpretations. We relate our observations to relaxation currents given via Curie-von Schweidler and Kohlraush laws. The Curie-von Schweidler law gives singular function power law as basic relaxation current in dielectric relaxation; whereas the Kohlraush law is Electric field relaxation in dielectric as non-Debye function taken as stretched exponential i.e. non-singular function. These two laws are used since late nineteenth century for various dielectric relaxation and characterization studies. Here we arrive at general constitutive equation for capacitor and with each type of memory kernel we give corresponding impedance function and in some cases equivalent circuit representation for the capacitor element. We classify these systems as Curie-von Shweidler type for system with singular memory kernel function and Kohlraush type for system evolved via using non-singular function or mixed functions as memory kernel. We note that use of singular memory kernel gives constituent relations and impedance functions that are experimentally verified in large number of cases of dielectric studies. Therefore, we have a question, does natural relaxation dynamics for dielectrics have a singular memory kernel, and the relaxation current function is singular in nature? Is it the singular relaxation function for capacitor dynamics with singular memory kernel remains universal law for dielectric relaxation? However, we are not questioning researchers modeling relaxation of dielectric via non-singular functions, yet we are hinting about complexity and lack of interpretability of basic constituent equation of dielectric relaxation dynamics thus obtained via considering non-singular and mixed memory kernels; perhaps due to insufficient experimental evidences presently. However, the method employed in this study is general method. This method can be used to form memorized constituent equations for other systems (say Radioactive Decay/Growth, Diffusion and Wave phenomena) from basic evolution equation, i.e. effect function as convolution of memory kernel with cause function.

2020, 13(3): 609-627 doi: 10.3934/dcdss.2020033 +[Abstract](2580) +[HTML](685) +[PDF](347.87KB)
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We prove Hölder regularity results for nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved. Moreover, Hölder continuity of viscosity solutions is obtained.

2020, 13(3): 629-643 doi: 10.3934/dcdss.2020034 +[Abstract](2968) +[HTML](572) +[PDF](792.34KB)
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Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order \begin{document}$\alpha = 1$\end{document}) and this chaos remains in the fractional case (\begin{document}$\alpha = 0.9$\end{document}). Moreover the dynamics of the system change depending on the parameter \begin{document}$\alpha$\end{document}, leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.

2020, 13(3): 645-662 doi: 10.3934/dcdss.2020035 +[Abstract](2781) +[HTML](697) +[PDF](654.49KB)
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Traveling waves remain significant in Applied Sciences mostly because they involve the movement of energy carrier particles. In this paper, traveling waves described by a generalized system, the fractional variable order hyperbolic Liouville model is solved numerically by means of Crank-Nicholson scheme. Detailed analysis are performed and prove that the numerical method is stable and converges. Simulations reveal that the model's variable order derivative (a function of time and position variables) has a considerable impact on the dynamics of the whole system. It influences the movement and the shape of the resulting waves including their amplitude, their wavelength as well as their compression and rarefaction processes. Such a variable order derivative becomes, due to these results, a substantial parameter and non-constant tool for the regulation and control of models describing wave motion.

2020, 13(3): 663-682 doi: 10.3934/dcdss.2020036 +[Abstract](2915) +[HTML](770) +[PDF](880.92KB)
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Studying & understanding the bursting dynamics of membrane potential in neurobiology is captivating in applied sciences, with many features still to be uncovered. In this study, 2D and 3D neuronal activities given by models of Hindmarsh-Rose (HR) neurons with external current input are analyzed numerically with Haar wavelet method, proven to be convergent through error analysis. Our numerical analysis considers two control parameters: the external current \begin{document}$I^{\text{ext}}$\end{document} and the derivative order \begin{document}$\gamma$\end{document}, on top of the other seven usual parameters \begin{document}$a, b, c, d, \nu_1, \nu_2$\end{document} and \begin{document}$x_{\text{rest}}.$\end{document} Bifurcation scenarios for the model show existence of equilibria, both stable and unstable of type saddle and spiral. They also reveal existence of stable limit cycle toward which the trajectories get closer. Numerical approximations of solutions to the 2D model reveals that equilibria remains the same in all cases, irrespective of control parameters'values, but the observed repeated sequences of impulses increase as \begin{document}$\gamma$\end{document} decreases. This inverse proportionability reveals a system likely to be \begin{document}$\gamma-$\end{document}controlled with \begin{document}$\gamma$\end{document} varying from 1 down to 0. A similar observation is done for the 3D HR neuron model where regular burst is observed and turns into period-adding chaotic bifurcation (burst with uncountable peaks) as \begin{document}$\gamma$\end{document} changes from 1 down to 0.

2020, 13(3): 683-693 doi: 10.3934/dcdss.2020037 +[Abstract](3068) +[HTML](709) +[PDF](319.53KB)
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The objective of this paper is to study the unsteady rotational flow of some non Newtonian fluids with Caputo fractional derivative through an infinite circular cylinder by means of the finite Hankel and Laplace transform. The novelty of the work is that motion is produced by applying tangential force not a specific but general function of time on the boundary. Initially the cylinder is at rest and after time \begin{document}$t_{o} = 0^{+}$\end{document} it begins to rotate about its axis with an angular velocity \begin{document}$\tau_{o} g(t)$\end{document}. The obtained solutions of velocity field and shear stress have been presented under series form in terms of generalized \begin{document}$G$\end{document}-function, satisfying all imposed initial and boundary conditions. The corresponding solutions can be easily particularized to give similar solutions from existing literature for Oldroyd-B fluids, Maxwell fluids, Second grade fluids and Newtonian fluids with/without fractional derivatives performing similar motions.

2020, 13(3): 695-708 doi: 10.3934/dcdss.2020038 +[Abstract](2395) +[HTML](554) +[PDF](282.29KB)
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In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

2020, 13(3): 709-722 doi: 10.3934/dcdss.2020039 +[Abstract](5461) +[HTML](1451) +[PDF](306.74KB)
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In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

2020, 13(3): 723-739 doi: 10.3934/dcdss.2020040 +[Abstract](2734) +[HTML](995) +[PDF](352.96KB)
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In this paper, the existence, uniqueness and stability of random implicit fractional differential equations (RIFDs) with nonlocal condition and impulsive effect involving a generalized Hilfer fractional derivative (HFD) are discussed. The arguments are discussed via Krasnoselskii's fixed point theorems, Schaefer's fixed point theorems, Banach contraction principle and Ulam type stability. Some examples are included to ensure the abstract results.

2020, 13(3): 741-754 doi: 10.3934/dcdss.2020041 +[Abstract](2365) +[HTML](608) +[PDF](811.51KB)
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This paper presents a comparative study of fractional Fokker-Planck equations with various fractional derivative operators such as Caputo fractional derivative, Atangana-Baleanu fractional derivative and conformable fractional derivative. The new iterative method has been successively applied for finding approximate analytical solutions of the fractional Fokker-Planck equations with various fractional derivative operators. This method gives an analytical solution in the form of a convergent series with easily computable components. The behavior of solutions and the effects of different values of fractional order are shown graphically for various fractional derivative operators. Some examples are given to show ability of the method for solving the fractional Fokker-Planck equations.

2020, 13(3): 755-768 doi: 10.3934/dcdss.2020042 +[Abstract](3061) +[HTML](918) +[PDF](608.6KB)
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In this article, we extend the concept of triple Laplace transform to the solution of fractional order partial differential equations by using Caputo fractional derivative. The concerned transform is applicable to solve many classes of partial differential equations with fractional order derivatives and integrals. As a consequence, fractional order telegraph equation in two dimensions is investigated in detail and the solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. The same problem is also solved by taking into account the Atangana-Baleanu fractional derivative. Numerical plots are provided for the comparison of Caputo and Atangana-Baleanu fractional derivatives.

2020, 13(3): 769-779 doi: 10.3934/dcdss.2020043 +[Abstract](2101) +[HTML](813) +[PDF](322.78KB)
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The present article deals to study heat transfer analysis due to convection occurs in a fractionalized H2O-based CNTs nanofluids flowing through a vertical channel. The problem is modeled in terms of fractional partial differential equations using a modern trend of the fractional derivative of Atangana and Baleanu. The governing equation (momentum and energy equations) are subjected to physical initial and boundary conditions. The fractional Laplace transformation is used to obtain solutions in the transform domain. To obtain semi-analytical solutions for velocity and temperature distributions, the Zakian's algorithm is utilized for the Laplace inversions. For validation, the obtained solutions are compared in tabular form using Tzou's and Stehfest's numerical methods for Laplace inversion. The influence of fractional parameter is studied and presented in graphs and discussed.

2020, 13(3): 781-795 doi: 10.3934/dcdss.2020044 +[Abstract](2250) +[HTML](914) +[PDF](627.49KB)
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The model describing a prototype of an excitable system was extended using the newly established concept of fractional differential operators with non-local and non-singular kernel in this paper. We presented a detailed discussion underpinning the well-poseness of the extended model. Due to the non-linearity of the modified model, we solved it using a newly established numerical scheme for partial differential equations that combines the fundamental theorem of fractional calculus, the Laplace transform and the Lagrange interpolation approximation. We presented some numerical simulations that, of course reflect asymptotically the real world observed behaviors.

2020, 13(3): 797-804 doi: 10.3934/dcdss.2020045 +[Abstract](2168) +[HTML](594) +[PDF](322.57KB)
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In this paper, we introduce the \begin{document}$k$\end{document}-generalised fractional derivatives with three parameters which reduced to \begin{document}$k$\end{document}-fractional Hilfer derivatives and \begin{document}$k$\end{document}-Riemann-Liouville fractional derivative as an interesting special cases. Further, we have also introduced some presumably new fascinating results which include the image power function, Laplace transform and composition of \begin{document}$k$\end{document}-Riemann-Liouville fractional integral with generalized composite fractional derivative. The technique developed in this paper can be used in other situation as well.

2020, 13(3): 805-821 doi: 10.3934/dcdss.2020046 +[Abstract](2828) +[HTML](722) +[PDF](443.35KB)
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This paper presents the solution for a fractional Bergman's minimal blood glucose-insulin model expressed by Atangana-Baleanu-Caputo fractional order derivative and fractional conformable derivative in Liouville-Caputo sense. Applying homotopy analysis method and Laplace transform with homotopy polynomial we obtain analytical approximate solutions for both derivatives. Finally, some numerical simulations are carried out for illustrating the results obtained. In addition, the calculations involved in the modified homotopy analysis transform method are simple and straightforward.

2020, 13(3): 823-834 doi: 10.3934/dcdss.2020047 +[Abstract](2708) +[HTML](765) +[PDF](942.92KB)
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In this paper, the local derivative in time is replaced with the Caputo-Fabrizio fractional derivative of order \begin{document}$\alpha\in(0, 1)$\end{document}. A two-step fractional version of the Adams-Bashforth method is formulated for the approximation of this derivative. To enhance the correct choice of parameters when numerically simulating the full-system, we examine the stability analysis of the main equation. Two important examples are drawn to explore the dynamic richness of the predator-prey model with Holling type. Simulation results at different instances of \begin{document}$\alpha$\end{document} is in agreement with the theoretical findings.

2020, 13(3): 835-851 doi: 10.3934/dcdss.2020048 +[Abstract](3132) +[HTML](801) +[PDF](1276.22KB)
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Ginzburg-Landau equation has a rich record of success in describing a vast variety of nonlinear phenomena such as liquid crystals, superfluidity, Bose-Einstein condensation and superconductivity to mention a few. Fractional order equations provide an interesting bridge between the diffusion wave equation of mathematical physics and intuition generation, it is of interest to see if a similar generalization to fractional order can be useful here. Non-integer order partial differential equations describing the chaotic and spatiotemporal patterning of fractional Ginzburg-Landau problems, mostly defined on simple geometries like triangular domains, are considered in this paper. We realized through numerical experiments that the Ginzburg-Landau equation world is bounded between the limits where new phenomena and scenarios evolve, such as sink and source solutions (spiral patterns in 2D and filament-like structures in 3D), various core and wave instabilities, absolute instability versus nonlinear convective cases, competition and interaction between sources and chaos spatiotemporal states. For the numerical simulation of these kind of problems, spectral methods provide a fast and efficient approach.

2020, 13(3): 853-865 doi: 10.3934/dcdss.2020049 +[Abstract](2446) +[HTML](635) +[PDF](335.62KB)
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This paper deals with fractional input stability, and contributes to introducing a new stability notion in the stability analysis of fractional differential equations (FDEs) with exogenous inputs using the Caputo fractional derivative. In particular, we study the fractional input stability of FDEs with exogenous inputs. A Lyapunov characterization of this notion is proposed and several examples are provided to explain the fractional input stability of FDEs with exogenous inputs. The applicability and simulation of this method are illustrated by studying the particular class of fractional neutral networks.

2020, 13(3): 867-880 doi: 10.3934/dcdss.2020050 +[Abstract](2662) +[HTML](745) +[PDF](312.24KB)
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This paper addresses the Mittag-Leffler input stability of the fractional differential equations with exogenous inputs. We continuous the first note. We discuss three properties of the Mittag-Leffler input stability: converging-input converging-state, bounded-input bounded-state, and Mittag-Leffler stability of the unforced fractional differential equation. We present the Lyapunov characterization of the Mittag-Leffler input stability, and conclude by introducing the fractional input stability for delay fractional differential equations, and we provide its Lyapunov-Krasovskii characterization. Several examples are treated to highlight the Mittag-Leffler input stability.

2020, 13(3): 881-888 doi: 10.3934/dcdss.2020051 +[Abstract](1846) +[HTML](563) +[PDF](312.13KB)
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We introduced the fading memory effect to the model portraying the prediction in physical condition. The classical model is known as the Banister model. We presented the existence and uniqueness conditions of the exact solutions of this model using three different memory including the bad memory induces by the power law and the good memories induced by exponential decay law and the Mittag-Leffler law. We derived the exact solutions using the Laplace transform for the non-delay version.

2020, 13(3): 889-909 doi: 10.3934/dcdss.2020052 +[Abstract](2218) +[HTML](674) +[PDF](1598.86KB)
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Models at which not only the asset price but also the volatility are assumed to be stochastic have received a remarkable attention in financial markets. The objective of the current research is to design a numerical method for solving the stochastic volatility (SV) jump–diffusion model of Bates, at which the presence of a nonlocal integral makes the coding of numerical schemes intensive. A numerical implementation is furnished by gathering several different techniques such as the radial basis function (RBF) generated finite difference (FD) approach, which keeps the sparsity of the FD methods but gives rise to the higher accuracy of the RBF meshless methods. Computational experiments are worked out to reveal the efficacy of the new procedure.

2020, 13(3): 911-923 doi: 10.3934/dcdss.2020053 +[Abstract](3967) +[HTML](856) +[PDF](399.15KB)
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The paper is relevance with Hilfer derivative with fractional order which is generalized case of R-L and Caputo's sense. We ensured the solution using noncompact measure and M$\ddot{\text{o}}$nch's fixed point technique. Illustrative examples are included for the applicability of presented technique.

2020, 13(3): 925-936 doi: 10.3934/dcdss.2020054 +[Abstract](2410) +[HTML](904) +[PDF](1151.13KB)
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In this paper, we study the space-time fractional perturbed nonlinear Schr\begin{document}$\bf{\ddot o}$\end{document}dinger equation under the Kerr law nonlinearity by using the extended sinh-Gordon equation expansion method. The perturbed nonlinear Schr\begin{document}$\bf{\ddot o}$\end{document}dinger equation is a nonlinear model which arises in nano-fibers. Some family of optical solitons and singular periodic wave solutions are successfully revealed. The parametric conditions for the existence of valid solitons are stated. Under the choice of suitable values of the parameters, the 3-dimensional and 2-dimensional graphs to some of the reported solutions are plotted.

2020, 13(3): 937-956 doi: 10.3934/dcdss.2020055 +[Abstract](2689) +[HTML](695) +[PDF](454.67KB)
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In the present paper, we explore the dynamics of fractional tuberculosis model with Atangana-Baleanu (A-B) derivative. The number of confirmed notified cases reported by national tuberculosis control program (NTP) Khyber Pakhtunkhwa, Pakistan, since 2002 to 2017 are used for our analysis and estimation of the model parameters. Initially, the essential properties of the model are presented. We prove the existence of the solution through fixed-point theory. Then, we show the uniqueness of the solution. Modified Adams-Bashforth technique is used to obtain the numerical solution of the fractional model. We obtain numerical results with different values of the fractional order parameters to show the importance of the newly proposed derivative, which provides useful information about the TB dynamics and its control.

2020, 13(3): 957-974 doi: 10.3934/dcdss.2020056 +[Abstract](2831) +[HTML](1028) +[PDF](2481.32KB)
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Hepatitis B is a viral infection that can cause both acute and chronic disease and mainly attacks the liver. The present paper describes the dynamics of HBV with hospitalization. Due to the fatal nature of this disease, it is necessary to formulate a new mathematical model in order to reduce the burden of HBV. Therefore, we formulate a new HBV model with fractional order derivative. The fractional order model is formulated in Caputo sense. Two equilibria for the model exist: the disease-free and the endemic equilibriums. It is shown, that the disease-free equilibrium is both locally and globally asymptotically stable if \begin{document}$\mathcal{R}_0<1$\end{document} for any \begin{document}$\alpha\in(0,1)$\end{document}. The sensitivity analysis of the model parameters are calculated and their results are depicted. The numerical results for the stability of the endemic equilibrium are presented. The complex dynamics of the disease can be best described by using the fractional derivative and this is illustrated through many graphical results.

2020, 13(3): 975-993 doi: 10.3934/dcdss.2020057 +[Abstract](3187) +[HTML](1046) +[PDF](478.08KB)
Abstract:

In the present paper, we study the dynamics of tuberculosis model using fractional order derivative in Caputo-Fabrizio sense. The number of confirmed notified cases reported by national TB program Khyber Pakhtunkhwa, Pakistan, from the year 2002 to 2017 are used for our analysis and estimation of the model biological parameters. The threshold quantity \begin{document}$\mathcal{R}_0$\end{document} and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of the model variables. An iterative solution of the model is computed using fractional Adams-Bashforth technique. Finally, the numerical results are presented by using the estimated values of model parameters to justify the significance of the arbitrary fractional order derivative. The graphical results show that the fractional model of TB in Caputo-Fabrizio sense gives useful information about the complexity of the model and one can get reliable information about the model at any integer or non-integer case.

2020, 13(3): 995-1006 doi: 10.3934/dcdss.2020058 +[Abstract](3802) +[HTML](686) +[PDF](664.49KB)
Abstract:

In this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator by using exponential kernel. After one year, Atangana and Baleanu recommended a different-type fractional operator that uses the generalized Mittag-Leffler function (MLF). Many real-life problems can be modelled and can be solved by numerical-analytical solution methods which are derived with these operators. In this paper, we suggest an approximate solution method for PDEs of fractional order by using the mentioned operators. We consider the Laplace homotopy transformation method (LHTM) which is the combination of standard homotopy technique (SHT) and Laplace transformation method (LTM). In this study, we aim to demonstrate the effectiveness of the aforementioned method by comparing the solutions we have achieved with the exact solutions. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.

2020, 13(3): 1007-1015 doi: 10.3934/dcdss.2020059 +[Abstract](2187) +[HTML](630) +[PDF](313.47KB)
Abstract:

The aim of this article is to study the natural convection boundary-layer flow over a semi-infinite heated plate with arbitrary inclination. Existing solutions of similar models can be recovered as the limiting cases of horizontal and vertical plates from our generalized problem. Moreover, porous effects and the influence of transverse magnetic field; fixed to the fluid or the plate are accounted. The dimensionless velocity, in conjunction with the corresponding skin friction, have been presented as the sum of mechanical, thermal and concentration components. Furthermore, the contribution of the system parameters to the fluid motion in question has been depicted graphically. The novelty of the present study is to analyze the effect of angle of inclination of the plate and the case when the magnetic field is fixed relative to the fluid or to the plate on the fluid motion.

2020, 13(3): 1017-1029 doi: 10.3934/dcdss.2020060 +[Abstract](2359) +[HTML](574) +[PDF](316.82KB)
Abstract:

We consider the regional enlarged observability problem for fractional evolution differential equations involving Caputo derivatives. Using the Hilbert Uniqueness Method, we show that it is possible to rebuild the initial state between two prescribed functions only in an internal subregion of the whole domain. Finally, an example is provided to illustrate the theory.

2019  Impact Factor: 1.233