In the present paper, for the first time, we propose a variable-order hyperchaotic system for information security. Firstly, we study the dynamical behaviors of a memristor oscillator through well-known numerical and analytical tools, such as the Lyapunov exponents, stability of equilibria, and bifurcation diagram. Then as an engineering application, a variable-order fractional version of the system is proposed for sound encryption. In comparison with integer and conventional constant fractional-order chaotic memristor oscillator, the proposed variable-order fractional system shows more complex characteristics and more degrees of freedom due to the existence of time-varying fractional derivatives. Thus, the proposed system is an appropriate choice for data transmission and information security. To illustrate the proper performance of the suggested system for encryption purposes, sound encryption is successfully performed, and its excellent results are demonstrated. The predictor-corrector method is utilized for numerical simulation. Then, a new type-2 fuzzy disturbance observer-based robust control is offered for synchronization of the variable-order hyperchaotic system. The stability and convergence of the disturbance estimator and closed-loop system are proven. Lastly, the synchronization results, which confirm the appropriate performance of the proposed method in the presence of the external disturbances, are demonstrated.

On the variable-order fractional memristor oscillator: Data security applications and synchronization using a type-2 fuzzy disturbance observer-based robust control

In this novel research, through dynamical analysis, we introduce for the first time a fractional-calculus based artificial macroeconomic model, actually implemented in the Laboratory via a new hardware set up. Firstly, we propose a new model of a discrete-time macroeconomic system where fractional derivatives are incorporated into the system of equations. Using well-known tools and methods, including bifurcation diagrams and Lyapunov exponents, the characteristics of the system are disclosed, and the importance of the fractional-order derivative in the modeling of the system is shown. After that, a laboratory hardware realization is also carried out for the proposed system that provides further insight and a better understanding of the properties of the system. For the hardware realization an Arduino Due™ is chosen in which possess two analog output pins. Experimental results conspicuously illustrate the chaotic behavior of the system. Through the results of the hardware realization, phase portraits and bifurcation diagram of the system are demonstrated, and the effects of the parameters and fractional derivatives are studied. We believe the presented study and its results pave the way for future studies on the incorporation of fractional calculus into macroeconomic models.

Artificial macro-economics: A chaotic discrete-time fractional-order laboratory model

Since December 2019, the new coronavirus has raged in China and subsequently all over the world. From the first days, researchers have tried to discover vaccines to combat the epidemic. Several vaccines are now available as a result of the contributions of those researchers. As a matter of fact, the available vaccines should be used in effective and efficient manners to put the pandemic to an end. Hence, a major problem now is how to efficiently distribute these available vaccines among various components of the population. Using mathematical modeling and reinforcement learning control approaches, the present article aims to address this issue. To this end, a deterministic Susceptible-Exposed-Infectious-Recovered-type model with additional vaccine components is proposed. The proposed mathematical model can be used to simulate the consequences of vaccination policies. Then, the suppression of the outbreak is taken to account. The main objective is to reduce the effects of Covid-19 and its domino effects which stem from its spreading and progression. Therefore, to reach optimal policies, reinforcement learning optimal control is implemented, and four different optimal strategies are extracted. Demonstrating the efficacy of the proposed methods, finally, numerical simulations are presented.

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Application of reinforcement learning for effective vaccination strategies of coronavirus disease 2019 (COVID-19)

On the dynamical investigation and synchronization of variable-order fractional neural networks: the Hopfield-like neural network model

Error analysis of an implicit Galerkin meshfree scheme for general second-order parabolic problems

The stability and convergence of the Galerkin method for differential equations with symmetric operators have been confirmed with numerical results, while this is not the case when dealing with unsymmetric operators. In the present study, a sort of transformation is used as a preconditioner to convert the unsymmetric operator to a symmetric one. This method is implemented on the capillary formation mathematical model of tumor angiogenesis problem. Then, a Galerkin meshfree method based on the radial basis functions is presented for the numerical solution of this problem. The proposed strategy is based on applying the Galerkin method and group preserving scheme for the spatial and time variables, respectively. Also, the stability and the convergence of proposed method is considered. In addition, some of the advantages of the proposed technique over existing methods are shown. Finally, some numerical results will be provided to validate the theoretical achievements.

Numerical analysis of Galerkin meshless method for parabolic equations of tumor angiogenesis problem

The present paper aims to carry out a new scheme for solving a type of singularly perturbed boundary value problem with a second order delay differential equation. Getting through the solution, we used Reproducing Kernel Hilbert Space (RKHS) method as an efficient approach to obtain the analytical solution for ordinary or partial differential equations that appear in vast areas of science and engineering. A key of this method is to keeping the continuous form of problems. Indeed, without discretizing the continuous problem, we change it to an equivalent iterative form and proving its convergence. Also, we will present a construction of the reproducing kernel in Hilbert space that satisfying the homogeneous nonlinear boundary conditions of the considered problem. Accuracy amount of absolute error with respect to different parameters of singularity has been studied for the performance of this method by solving several hardly nonlinear problems. Error estimation and convergence analysis show that the approximate results have uniform convergence to the continuous solutions.

Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay

This paper introduces an efficient numerical scheme  for solving a significant class of  nonlinear parabolic integro-differential equations (PIDEs). The major contributions made in this paper are applying a direct approach based on a combination of   group preserving scheme (GPS) and  spectral meshless radial point interpolation (SMRPI) method   to transcribe the partial differential problem under study into a system of ordinary differential equations (ODEs). The resulting problem is then solved by employing the numerical method of lines, which is also a well-developed numerical method.   Two numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

https://jmm.guilan.ac.ir/article_4037_6fc65daf4904bd5ada4c05abbf37e869.pdf

The method of lines for parabolic integro-differential equations

Radial basis solutions of second‐order quasi‐linear hyperbolic boundary value problem

This paper aims to investigate the stability and numerical approximation of the Sivashinsky equations. We apply the Galerkin meshfree method based on the radial basis functions (RBFs) to discretize the spatial variables and use a group presenting scheme for the time discretization. Because the RBFs do not generally vanish on the boundary, they can not directly approximate a Dirichlet boundary problem by Galerkin method. To avoid this difficulty, an auxiliary parametrized technique is used to convert a Dirichlet boundary condition to a Robin one. In addition, we extend a stability theorem on the higher order elliptic equations such as the biharmonic equation by the eigenfunction expansion.Some experimental results will be presented to show the performance of the proposed method.

https://cmde.tabrizu.ac.ir/article_9208_7402408cf46500e9f5e2f568e39b4c66.pdf

Mehdi Mesrizadeh

Papers

Numerical analysis of Galerkin meshless method for parabolic equations of tumor angiogenesis problem

Reproducing kernel Hilbert space method for nonlinear second order singularly perturbed boundary value problems with time-delay

The method of lines for parabolic integro-differential equations

Radial basis solutions of second‐order quasi‐linear hyperbolic boundary value problem

Stability and numerical approximation for a spacial class of semilinear parabolic equations on the Lipschitz bounded regions: Sivashinsky equation