Volume 24, Issue 7

In this paper, we classify G -circuits of length 10 with the help of the location of the reduced numbers lying on G -circuit. The reduced numbers play an important role in the study of modular group action on P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -subset of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ . For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G -circuits in P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of real quadratic fields. In particular, we classify P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits of Q ( m ) \ Q $Q(\sqrt{m}){\backslash}Q$ = ⋃ k ∈ N Q * k 2 m ${=}{\bigcup }_{k\in N}{Q}^{\ast }\left(\sqrt{{k}^{2}m}\right)$ containing G -circuits of length 10 and determine that the number of equivalence classes of G -circuits of length 10 is 41 in number. We also use dihedral group to explore cyclically equivalence classes of circuits and use cyclic group to explore similar G -circuits of length 10 corresponding to 10 of these circuits. By using cyclically equivalent classes of circuits and similar circuits, we obtain the exact number of G -orbits and the structure of G -circuits corresponding to cyclically equivalent classes. This study also helps us in classifying the reduced numbers lying in the P S L ( 2 , Z ) $PSL(2,\mathbb{Z})$ -orbits.

In this paper, we study a new kind of generalized Caputo fractional differential quasi-variational inequalities in Hilbert spaces. We prove the uniqueness and the stability of the abstract inequality by using generalized singular Gronwall’s lemma, projection operators, and contraction principle. Finally, an example is given to illustrate the abstract results.

Synchronization phenomena are related to several natural systems defining patterns of interactions. This paper deals with a synchronization robustness investigation evaluating pattern formation on a network of pendulum-chart oscillators receiving energy from a base excitation. Random aspects are investigated establishing the sensitivity to parameter changes and initial conditions. System asymmetries are analysed allowing the investigation of different kinds of dynamical patterns. Results show that asynchronous regions can change due to random effects. The asynchronous region reduces with the dissipation increase and the chimera state can occur under parametric asymmetry. Energetic argues are employed to explain the pattern robustness with respect to randomness and asymmetries.

In this study, we present a mathematical model of socio-economic process of labor force migration. We present our model in the terms of distributed order fractional partial differential equation. The proposed model involves time varying diffusion coefficient. To solve this model we propose an implicit discrete scheme by using graded meshes, and study numerically labor migration flow data from Turkey along with other countries throughout the period of 1966–2014.

In this typescript, we study system of nonlinear implicit coupled differential equations of arbitrary (non–integer) order having nonlocal boundary conditions on closed interval [0, 1] with Caputo fractional derivative. We establish sufficient conditions for the existence, at least one and a unique solution of the proposed coupled system with the help of Krasnoselskii’s fixed point theorem and Banach contraction principle. Moreover, we scrutinize the Hyers–Ulam stability for the considered problem. We present examples to illustrate our main results.

This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

In this paper, a generalized stochastic SIR epidemic model with vaccination rules is presented and the threshold behavior of the proposed epidemic model is investigated. Firstly, the stability of the equilibrium of the deterministic system is considered and the corresponding conditions are obtained. Secondly, the threshold of a stochastic SIR system for the extinction and the permanence in mean of epidemic disease are investigated. The results show that a larger stochastic disturbance can cause infections diseases to go to extinction. However, for a relatively small stochastic disturbance, the evolutionary dynamics of the epidemic diseases are overwhelmingly depend on the incidence function. This implies that the stochastic disturbance and the incidence function play an important role in diseases control. To test the theoretical results, a series of numerical simulations of these cases with respect to different noise disturbance coefficients are conducted.

A detailed survey of existing works on fractional-order nonlinear systems reveals the fact that practically no results exist on stability or any performance analysis of Markovian jumping fractional-order systems (FOSs) in general. The main reason is the theory of infinitesimal generator used to estimate the derivative of Lyapunov–Krasovskii Functional (LKF) is not well-developed in the fractional domain. This shortage, in theory, is focussed in this manuscript. In this work, we provide a lemma that aids in analyzing the stability of fractional-order delayed systems via integer-order derivative of LKF. Using this lemma, by constructing a new suitable LKF and employing known integral inequalities, linear matrix inequality (LMI)-based sufficient conditions that ensure stability along with H ∞ /passive performance of the proposed fractional-order neural networks (FONNs) with Markovian jumping parameters are derived for the first time. Unlike the existing works, the results derived in the present study depend on the fractional order (FO) of the NNs. The importance of such order-dependent criteria is highlighted in numerical examples. Finally, the simulation results are given to show the reliability of the derived conditions.

The penetration efficiency of novel ceramic fragments should be investigated, and their weapon damage effectiveness must be evaluated. In this study, the efficiency of ceramic fragments in penetrating steel targets were analyzed through ballistic impact tests and numerical simulations. The penetration patterns of these ceramic fragments through steel targets indicate significant perforation. It was deduced that the thicker the steel target, the greater the ability of ZrO 2 ceramic fragments to penetrate. Results indicate that the thicker the steel target, the greater the ability of ZrO 2 ceramic fragments to expand their perforation, while that of Al 2 O 3 ceramic fragments is reduced as the thickness of the steel target is increased. In addition, the number of projectiles triggered by the perforation of the two ceramic fragments behind the targets decreases as the thickness of the steel target increases. Thus, the higher the impact velocity, the larger the perforation diameter of the ceramic fragments, and the larger the number of projectiles behind the target. Under the same impact velocity, the penetration ability of a ZrO 2 ceramic fragment on steel targets was better than that of an Al 2 O 3 ceramic fragment, and the number of projectiles behind the target was greater.

We discuss the existence of solutions for a boundary value problem of nonlinear coupled Riemann–Liouville fractional integro-differential equations equipped with coupled nonlocal fractional integro-multipoint boundary conditions. The standard tools of the modern functional analysis are employed to derive the desired results for the problem at hand. The case of nonlinearities depending on the Riemann–Liouville fractional integrals is also discussed. Examples illustrating the obtained results are presented.

The article deals with the approximate controllability of Atangana–Baleanu semilinear control systems. The outcomes are derived by applying Gronwall’s inequality and Cauchy sequence, and avoid the use of the fixed point theorem. We have also included an example for the validation of theoretical results.

This paper discusses the complex dynamic behavior of a novel chaotic system, which was firstly established by introducing a memristor into a similar Chen’s system. Then by choosing a as the key parameter, we analyze the stability of memristor system based on eigenvalue theory. It is also found that when a cross some critical values, the system can exhibit Neimark–Sacker bifurcation and chaos behaviors. Some numerical simulations including phase diagrams and maximum Lyapunov exponent graph of the memristor-based systems are presented to verify the existence of chaos attractors. Finally, to make the results of this paper useful in the actual situation, such as the design of chaos security algorithm, analog electronic circuit of memristor chaotic system is designed.

In this article, we investigate the delay-dependent and order-dependent dissipativity analysis for a class of Caputo fractional-order neural networks (FONNs) subject to time-varying delays. By employing the Razumikhin fractional-order (RFO) approach combined with linear matrix inequalities (LMIs) techniques, a new sufficient condition is derived to guarantee that the considered fractional-order is strictly (Q, S, R) − γ − dissipativity. The condition is presented via LMIs and can be efficiently checked. Two numerical examples and simulation results are finally provided to express the effectiveness of the obtained results.

A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability.

In this note, the main emphasis is to study two kinds of high-order fractional p -Laplacian differential equations with mixed derivatives, which include Caputo type and Riemann–Liouville type fractional derivative. Based on fixed point theorems on the cone, the existence-uniqueness of positive solutions for equations and two iterative schemes to uniformly approximate the unique solutions are discussed theoretically. What’s more, the sufficient conditions for stability of the equations are given. Some exact examples are further provided to verify the analytical results at the end of the article.

In this paper, we will solve a nonlinear time-fractional equation with nonlocal boundary conditions. First, we construct a set of suitable base according to the correlation theory of reproducing kernel space and the nonlocal boundary conditions. Then we introduce F -derivative and Newton iterative to linearize the nonlinear terms of the equation. The appropriate initial iteration value is constructed by boundary conditions. Finally, we use the ɛ -approximate solution theory to solve the system of linear equations and give the corresponding theoretical derivation. In conclusion, several numerical examples illustrate the feasibility and effectiveness of the method.

In this paper, the global asymptotic stability (GAS) of continuous-time and discrete-time nonlinear impulsive switched positive systems (NISPS) are studied. For continuous-time and discrete-time NISPS, switching signals and impulse signals coexist. For both of these systems, using the multiple max-separable Lyapunov function method and average dwell-time (ADT) method, some sufficient conditions on GAS are given. Based on these, the GAS criteria are also given for continuous-time and discrete-time linear impulsive switched positive systems (LISPS). From our criteria, the stability of the systems can be judged directly from the characteristics of the system functions, switching signals and impulse signals of the systems. Finally, simulation examples verify the validity of the results.

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

The aim of this work is to modify the invariant subspace method (ISM) in order to obtain closed form solutions of fractional differential-difference equations with Ψ-Riemann–Liouville (Ψ-RL) fractional derivative for first time. We have investigated the cases of two-dimensional and the three-dimensional invariant subspaces (ISs) in the suggested scheme. Using the modified ISM, new exact generalized solutions for the general fractional mKdV Lattice equation and the fractional Volterra lattice system are obtained. Compared with similar solution techniques in literature, the presented solution scheme is highly efficient and is capable to find new general exact solutions which cannot be attained by other methods.

We propose a novel framework for model-order reduction of hyperbolic differential equations. The approach combines a relaxation formulation of the hyperbolic equations with a discretization using shifted base functions. Model-order reduction techniques are then applied to the resulting system of coupled ordinary differential equations. On computational examples including in particular the case of shock waves we show the validity of the approach and the performance of the reduced system.

In this article, we introduce a new Steffensen-type method with the advantage that its behavior is very similar to Newton’s method; therefore, it is a very remarkable way of avoiding the drawback that Newton’s method presents for nondifferentiable operators. In our study, we perform an exhaustive comparative study between the semilocal convergence of Newton’s method and the derivative-free point-to-point iterative process considered; in the case of differentiable operators, we use the majoring sequences and the majorant principle. In the nondifferentiable case, we impose conditions on the starting point and on the nonlinear operator to obtain a semilocal convergence result for the iterative process considered. In both cases, we complete the theoretical convergence proofs with a dynamical study and a numerical test. In the case of differentiable operators, this study confirms that the accessibility and numerical behavior of both iterative processes, Newton’s method and the derivative-free point-to-point iterative process considered, are very similar.

In this paper, we investigate a newly introduced integrable (2 + 1)-dimensional Boussinesq equation. Solutions of this equation are obtained by Lie symmetry reductions and direct integration. We achieve diverse solitary wave solutions of the equation among which are non-topological soliton as well as Jacobi elliptic function solutions. Moreover, we generate some closed-form solutions of the equation which are in the form of bright, singular and non-singular periodic solitons. Power series solution of the equation is also generated. In a bid to have a sound understanding of the physical phenomena of the underlying model, we exhibited graphically the motion of the secured results. Besides, we discuss the obtained results as well as their respective graphs. Conclusively, we construct conservation laws of the aforementioned equation by employing the general multiplier approach.