# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

October 2021 , Volume 14 , Issue 10

Issue on recent advances in nonlinear dynamics and modeling

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2021, 14(10): i-i doi: 10.3934/dcdss.2021094 +[Abstract](112) +[HTML](35) +[PDF](70.3KB)
Abstract:
2021, 14(10): 3337-3349 doi: 10.3934/dcdss.2020443 +[Abstract](973) +[HTML](363) +[PDF](338.1KB)
Abstract:

In this work, we examine the oscillation of a class fractional differential equations in the frame of generalized nonlocal fractional derivatives with function dependent kernel type. We present sufficient conditions to prove the oscillation criteria in both of the Riemann-Liouville (RL) and Caputo types. Taking particular cases of the nondecreasing function appearing in the kernel of the treated fractional derivative recovers the oscillation of several proven results investigated previously in literature. Two examples, where the kernel function is quadratic and cubic polynomial, have been given to support the validity of the proven results for the RL and Caputo cases, respectively.

2021, 14(10): 3351-3386 doi: 10.3934/dcdss.2020440 +[Abstract](1253) +[HTML](366) +[PDF](470.06KB)
Abstract:

Crisis intervention in natural disasters is significant to look at from many different slants. In the current study, we investigate the existence of solutions for \begin{document}$q$\end{document}-integro-differential equation

with three criteria and under some boundary conditions which therein we use the concept of Caputo fractional \begin{document}$q$\end{document}-derivative and fractional Riemann-Liouville type \begin{document}$q$\end{document}-integral. New existence results are obtained by applying \begin{document}$\alpha$\end{document}-admissible map. Lastly, we present two examples illustrating the primary effects.

2021, 14(10): 3387-3399 doi: 10.3934/dcdss.2021017 +[Abstract](907) +[HTML](403) +[PDF](279.61KB)
Abstract:

The aim of this paper is to study the calcium profile governed by the advection diffusion equation. The mathematical and computational modeling has provided insights to understand the calcium signalling which depends upon cytosolic calcium concentration. Here the model includes the important physiological parameters like diffusion coefficient, flow velocity etc. The mathematical model is fractionalised using Hilfer derivative and appropriate boundary conditions have been framed. The use of fractional order derivative is more advantageous than the integer order because of the non-local property of the fractional order differentiation operator i.e. the next state of the system depends not only upon its current state but also upon all of its preceeding states. Analytic solution of the fractional advection diffusion equation arising in study of diffusion of cytosolic calcium in RBC is found using integral transform techniques. Since, the Hilfer derivative is generalisation of Riemann- Liouville and Caputo derivatives so, these two are also deduced as special cases. The numerical simulation has been done to observe the effects of the fractional order of the derivatives involved in the differential equation representing the model over the concentration of calcium which is function of time and distance. The concentration profile of calcium is significantly changed by the fractional order.

2021, 14(10): 3401-3417 doi: 10.3934/dcdss.2020423 +[Abstract](1201) +[HTML](450) +[PDF](911.44KB)
Abstract:

We obtain the solutions of fractal fractional differential equations with the power law kernel by reproducing kernel Hilbert space method in this paper. We also apply the Laplace transform to get the exact solutions of the problems. We compare the exact solutions with the approximate solutions. We demonstrate our results by some tables and figures. We prove the efficiency of the proposed technique for fractal fractional differential equations.

2021, 14(10): 3419-3440 doi: 10.3934/dcdss.2020426 +[Abstract](1241) +[HTML](444) +[PDF](587.86KB)
Abstract:

Prostate cancer worldwide is regarded the second most frequent diagnosed cancer in men with (899,000 new cases) while in common cancer it is the fifth. Regarding the treatment of progressive prostate cancer the most common and effective is the intermittent androgen deprivation therapy. Usually this treatment is effective initially at regressing tumorigenesis, mostly a resistance to treatment can been seen from patients and is known as the castration-resistant prostate cancer (CRPC), so there is no any treatment and becomes fatal. Therefore, we proposed a new mathematical model for the prostate cancer growth with fractional derivative. Initially, we present the model formulation in detail and then apply the fractional operator Atangana-Baleanu to the model. The fractional model will be studied further to analyze and show its existence of solution. Then, we provide a new iterative scheme for the numerical solution of the prostate cancer growth model. The analytical results are validated by considering various values assigned to the fractional order parameter \begin{document}$\alpha.$\end{document}

2021, 14(10): 3441-3457 doi: 10.3934/dcdss.2020421 +[Abstract](1939) +[HTML](577) +[PDF](1381.16KB)
Abstract:

New class of differential and integral operators with fractional order and fractal dimension have been introduced very recently and gave birth to new class of differential and integral equations. In this paper, we derive exact solution of some important ordinary differential equations where the differential operators are the fractal-fractional. We presented a new numerical scheme to obtain solution in the nonlinear case. We presented the numerical simulation for different values of fractional orders and fractal dimension.

2021, 14(10): 3459-3478 doi: 10.3934/dcdss.2021018 +[Abstract](1184) +[HTML](310) +[PDF](7764.42KB)
Abstract:

In this research paper, the modified Khater method, the Adomian decomposition method, and B-spline techniques (cubic, quintic, and septic) are applied to the deoxyribonucleic acid (DNA) model to get the analytical, semi-analytical, and numerical solutions. These solutions comprise much information about the dynamical behavior of the homogenous long elastic rods with a circular section. These rods constitute a pair of the polynucleotide rods of the DNA molecule which are plugged by an elastic diaphragm that demonstrates the hydrogen bond's role in this communication. The stability property is checked for some solutions to show more effective and powerful of obtained solutions. Based on the role of analytical and semi-analytical techniques in the motivation of the numerical techniques to be more accurate, the B-spline numerical techniques are applied by using the obtained exact solutions on the DNA model to show which one of them is more accurate than other, to explain more of the dynamic behavior of the homogenous long elastic rods, and to show the coincidence between the different types of obtained solutions. The obtained solutions verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows the power and effectiveness of them for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.

2021, 14(10): 3479-3495 doi: 10.3934/dcdss.2020425 +[Abstract](1077) +[HTML](426) +[PDF](407.47KB)
Abstract:

The present paper deals with the existence and multiplicity of solutions for a class of fractional \begin{document}$p(x,.)$\end{document}-Laplacian problems with the nonlocal Dirichlet boundary data, where the nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. The main results are established by means of mountain pass theorem and Fountain theorem with Cerami condition.

2021, 14(10): 3497-3528 doi: 10.3934/dcdss.2020442 +[Abstract](1087) +[HTML](387) +[PDF](478.95KB)
Abstract:

This paper studies a class of fourth point singular boundary value problem of \begin{document}$p$\end{document}-Laplacian operator in the setting of a specific kind of conformable derivatives. By using the upper and lower solutions method and fixed point theorems on cones., necessary and sufficient conditions for the existence of positive solutions are obtained. In addition, we investigate the dependence of the solution on the order of the conformable differential equation and on the initial conditions.

2021, 14(10): 3529-3539 doi: 10.3934/dcdss.2020432 +[Abstract](1130) +[HTML](371) +[PDF](362.66KB)
Abstract:

By a fuzzy controller function, we stable a random operator associated with a type of fractional stochastic Volterra integral equations. Using the fixed point technique, we get an approximation for the mentioned random operator by a solution of the fractional stochastic Volterra integral equation.

2021, 14(10): 3541-3556 doi: 10.3934/dcdss.2020441 +[Abstract](961) +[HTML](347) +[PDF](843.02KB)
Abstract:

This paper studies an \begin{document}$(n+2)$\end{document}-dimensional nonlinear HIV dynamics model that characterizes the interactions of HIV particles, susceptible CD4\begin{document}$^{+}$\end{document} T cells and \begin{document}$n$\end{document}-stages of infected CD4\begin{document}$^{+}$\end{document} T cells. Both virus-to-cell and cell-to-cell infection modes have been incorporated into the model. The incidence rates of viral and cellular infection as well as the production and death rates of all compartments are modeled by general nonlinear functions. We have revealed that the solutions of the system are nonnegative and bounded, which ensures the well-posedness of the proposed model. The basic reproduction number \begin{document}$\Re_{0}$\end{document} is determined which insures the existence of the two equilibria of the model under consideration. A set of conditions on the general functions has been established which is sufficient to investigate the global stability of the model's equilibria. The global asymptotic stability of the two equilibria is proven by utilizing Lyapunov function and LaSalle's invariance principle. We have proven that if \begin{document}$\Re_{0}\leq1$\end{document}, then the infection-free equilibrium is globally asymptotically stable, and if \begin{document}$\Re _{0}>1$\end{document}, then the chronic-infection equilibrium is globally asymptotically stable. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions.

2021, 14(10): 3557-3575 doi: 10.3934/dcdss.2020429 +[Abstract](1092) +[HTML](417) +[PDF](2883.92KB)
Abstract:

The present work explore the dynamics of the cancer model with fractional derivative. The model is formulated in fractional type of Caputo-Fabrizio derivative. We analyze the chaotic behavior of the proposed model with the suggested parameters. Stability results for the fixed points are shown. A numerical scheme is implemented to obtain the graphical results in the sense of Caputo-Fabrizio derivative with various values of the fractional order parameter. Further, we show the graphical results in order to study that the model behave the periodic and quasi periodic limit cycles as well as chaotic behavior for the given set of parameters.

2021, 14(10): 3577-3587 doi: 10.3934/dcdss.2020428 +[Abstract](1008) +[HTML](464) +[PDF](688.48KB)
Abstract:

The principal aim of the present article is to study a mathematical pattern of interacting phytoplankton species. The considered model involves a fractional derivative which enjoys a nonlocal and nonsingular kernel. We first show that the problem has a solution, then the proof of the uniqueness is included by means of the fixed point theory. The numerical solution of the mathematical model is also obtained by employing an efficient numerical scheme. From numerical simulations, one can see that this is a very efficient method and provides precise and outstanding results.

2021, 14(10): 3589-3610 doi: 10.3934/dcdss.2021021 +[Abstract](1116) +[HTML](344) +[PDF](1031.21KB)
Abstract:

In this paper, an effective analytical scheme based on Sumudu transform known as homotopy perturbation Sumudu transform method (HPSTM) is employed to find numerical solutions of time fractional Schrödinger equations with harmonic oscillator.These nonlinear time fractional Schrödinger equations describe the various phenomena in physics such as motion of quantum oscillator, lattice vibration, propagation of electromagnetic waves, fluid flow, etc. The main objective of this study is to show the effectiveness of HPSTM, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results reveal that proposed scheme is a powerful tool for study large class of problems. This study shows that the results obtained by the HPSTM are accurate and effective for analysis the nonlinear behaviour of complex systems and efficient over other available analytical schemes.

2021, 14(10): 3611-3628 doi: 10.3934/dcdss.2020431 +[Abstract](1090) +[HTML](367) +[PDF](4277.53KB)
Abstract:

The intension of the recent study is to solve a class of biological nonlinear HIV infection model of latently infected CD4+T cells using feed-forward artificial neural networks, optimized with global search method, i.e. particle swarm optimization (PSO) and quick local search method, i.e. interior-point algorithms (IPA). An unsupervised error function is made based on the differential equations and initial conditions of the HIV infection model represented with latently infected CD4+T cells. For the correctness and reliability of the present scheme, comparison is made of the present results with the Adams numerical results. Moreover, statistical measures based on mean absolute deviation, Theil's inequality coefficient as well as root mean square error demonstrates the effectiveness, applicability and convergence of the designed scheme.

2021, 14(10): 3629-3640 doi: 10.3934/dcdss.2020420 +[Abstract](1115) +[HTML](473) +[PDF](361.42KB)
Abstract:

In this paper, we introduce the notion of rational Pata type contraction in the complete metric space. After discussing the existence and uniqueness of a fixed point for such contraction, we consider a solution for integral equations.

2021, 14(10): 3641-3657 doi: 10.3934/dcdss.2020434 +[Abstract](1585) +[HTML](480) +[PDF](380.68KB)
Abstract:

This paper studies the problem of stabilization of some coupled hyperbolic systems using nonlinear feedback. We give a sufficient condition for exponential stabilization by bilinear feedback control. The specificity of the control used is that it acts on only one equation. The results obtained are illustrated by some examples where a theorem of Mehrenberger has been used for the observability of compactly perturbed systems [18].

2021, 14(10): 3659-3683 doi: 10.3934/dcdss.2021023 +[Abstract](1148) +[HTML](346) +[PDF](481.97KB)
Abstract:

This paper is concerned with the asymptotic behaviors of solution to time–space fractional partial differential equation with Caputo–Hadamard derivative (in time) and fractional Laplacian (in space) in the hyperbolic case, that is, the Caputo–Hadamard derivative order \begin{document}$\alpha$\end{document} lies in \begin{document}$1<\alpha<2$\end{document}. In view of the technique of integral transforms, the fundamental solutions and the exact solution of the considered equation are derived. Furthermore, the fundamental solutions are estimated and asymptotic behaviors of its analytical solution is established in \begin{document}$L^{p}(\mathbb{R}^{d})$\end{document} and \begin{document}$L^{p,\infty} (\mathbb{R}^{d})$\end{document}. We finally investigate gradient estimates and large time behavior for the solution.

2021, 14(10): 3685-3701 doi: 10.3934/dcdss.2020466 +[Abstract](1110) +[HTML](478) +[PDF](1077.66KB)
Abstract:

This paper aimed at obtaining the traveling-wave solution of the nonlinear time fractional regularized long-wave equation. In this approach, firstly, the time fractional derivative is accomplished using a finite difference with convergence order \begin{document}$\mathcal{O}(\delta t^{2-\alpha})$\end{document} for \begin{document}$0 < \alpha< 1$\end{document} and the nonlinear term is linearized by a linearization technique. Then, the spatial terms are approximated with the help of the radial basis function (RBF). Numerical stability of the method is analyzed by applying the Von-Neumann linear stability analysis. Three invariant quantities corresponding to mass, momentum and energy are evaluated for further validation. Numerical results demonstrate the accuracy and validity of the proposed method.

2021, 14(10): 3703-3718 doi: 10.3934/dcdss.2021020 +[Abstract](922) +[HTML](463) +[PDF](396.1KB)
Abstract:

The present article deals with the new estimates in the view of generalized proportional fractional integral with respect to another function. In the present investigation, we focus on driving certain new classes of integral inequalities utilizing a family of positive functions \begin{document}$n(n\in\mathbb{N})$\end{document} for this newly defined operator. From the computed outcomes, we concluded some new variants for classical generalized proportional fractional and other integrals as remarks. These variants are connected with some existing results in the literature. Certain interesting consequent results of the main theorems are also pointed out.

2021, 14(10): 3719-3746 doi: 10.3934/dcdss.2020430 +[Abstract](1292) +[HTML](574) +[PDF](5394.63KB)
Abstract:

This article is focused on the slip effect in the unsteady flow of MHD Oldroyd-B fluid over a moving vertical plate with velocity \begin{document}$U_{o}f(t)$\end{document}. The Laplace transformation and inversion algorithm are used to evaluate the expression for fluid velocity and shear stress. Fractional time derivatives are used to analyze the impact of fractional parameters (memory effect) on the dynamics of the fluid. While making a comparison, it is observed that the fractional-order model is best to explain the memory effect as compared to the classical model. The behavior of slip condition as well as no-slip condition is discussed with all physical quantities. The influence of dimensionless physical parameters like magnetic force \begin{document}$M$\end{document}, retardation time \begin{document}$\lambda_{r}$\end{document}, fractional parameter \begin{document}$\alpha$\end{document}, and relaxation time \begin{document}$\lambda$\end{document} on fluid velocity has been discussed through graphical illustration. Our results suggest that the velocity field decreases by increasing the value of the magnetic field. In the absence of a slip parameter, the strength of the magnetic field is maximum. Furthermore, it is noted that the Atangana-Baleanu derivative in Caputo sense (ABC) is the best to highlight the dynamics of the fluid.

2021, 14(10): 3747-3761 doi: 10.3934/dcdss.2020435 +[Abstract](1137) +[HTML](457) +[PDF](1133.34KB)
Abstract:

In recent years, a new definition of fractional derivative which has a nonlocal and non-singular kernel has been proposed by Atangana and Baleanu. This new definition is called the Atangana-Baleanu derivative. In this paper, we present a new technique to obtain the numerical solution of advection-diffusion equation containing Atangana-Baleanu derivative. For this purpose, we use the operational matrix of fractional integral based on Genocchi polynomials. An error bound is given for the approximation of a bivariate function using Genocchi polynomials. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method.

2021, 14(10): 3763-3783 doi: 10.3934/dcdss.2021022 +[Abstract](1001) +[HTML](362) +[PDF](526.05KB)
Abstract:

This study outlines a modiﬁed implicit ﬁnite difference method for approximating the local stable manifold near a hyperbolic equilibrium point for a nonlinear systems of fractional differential equations. The fractional derivative is described in the Caputo sense of order \begin{document}$\alpha\; (0<\alpha \le1)$\end{document} which is approximated based on the modified trapezoidal quadrature rule of order \begin{document}$O(\triangle t ^{2-\alpha})$\end{document}. The solution existence, uniqueness and stability of the proposed method is discussed. Three numerical examples are presented and comparisons are made to conﬁrm the reliability and effectiveness of the proposed method.

2021, 14(10): 3785-3801 doi: 10.3934/dcdss.2020433 +[Abstract](1198) +[HTML](373) +[PDF](580.72KB)
Abstract:

The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.

When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.

2021, 14(10): 3803-3819 doi: 10.3934/dcdss.2021019 +[Abstract](768) +[HTML](268) +[PDF](370.28KB)
Abstract:

In this article, we have investigated certain definite integrals and various integral transforms of the generalized multi-index Bessel function, such as Euler transform, Laplace transform, Whittaker transform, K-transform and Fourier transforms. Also found the applications of the problem on fractional kinetic equation pertaining to the generalized multi-index Bessel function using the Sumudu transform technique. Mittage-Leffler function is used to express the results of the solutions of fractional kinetic equation as well as its special cases. The results obtained are significance in applied problems of science, engineering and technology.

2021, 14(10): 3821-3836 doi: 10.3934/dcdss.2020436 +[Abstract](995) +[HTML](396) +[PDF](352.4KB)
Abstract:

In this paper, we study the positive solutions of the Schrödinger elliptic system

where \begin{document}$\mathcal{G}$\end{document} is a nonlinear operator. By using the monotone iterative technique and Arzela-Ascoli theorem, we prove that the system has the positive entire bounded radial solutions. Then, we establish the results for the existence and nonexistence of the positive entire blow-up radial solutions for the nonlinear Schrödinger elliptic system involving a nonlinear operator. Finally, three examples are given to illustrate our results.

2021, 14(10): 3837-3849 doi: 10.3934/dcdss.2020444 +[Abstract](1450) +[HTML](361) +[PDF](314.46KB)
Abstract:

In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of conformable Laplace transform.

2021, 14(10): 3851-3863 doi: 10.3934/dcdss.2020445 +[Abstract](1021) +[HTML](402) +[PDF](378.54KB)
Abstract:

In this paper, we investigate radial symmetry and monotonicity of positive solutions to a logarithmic Choquard equation involving a generalized nonlinear tempered fractional \begin{document}$p$\end{document}-Laplacian operator by applying the direct method of moving planes. We first introduce a new kind of tempered fractional \begin{document}$p$\end{document}-Laplacian \begin{document}$(-\Delta-\lambda_{f})_{p}^{s}$\end{document} based on tempered fractional Laplacian \begin{document}$(\Delta+\lambda)^{\beta/2}$\end{document}, which was originally defined in [3] by Deng et.al [Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul., 16(1)(2018), 125-149]. Then we discuss the decay of solutions at infinity and narrow region principle, which play a key role in obtaining the main result by the process of moving planes.

2020 Impact Factor: 2.425
5 Year Impact Factor: 1.490
2020 CiteScore: 3.1