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This study uses numerical methods to solve a specific type of reaction-diffusion problem arising in viscoelasticity (singularly perturbed Fredholm integro-differential equations). These equations are challenging because they exhibit ‘boundary layers’ near the edges of the area of interest. To approximate solutions, a technique known as a second-order scheme is used for derivatives, and the trapezoidal rule is used for integral terms. This is done on non-standard grids known as Shishkin-type meshes. We found that this numerical method and its rate of improvement (or convergence) are both of the second order, which means they improve consistently as the calculations continue. This improvement rate remains consistent even when dealing with small parameters in the equations. In addition, a post-processing method is used to enhance the rate of convergence from second order to almost fourth order, indicating a significant improvement in the speed and accuracy of the solutions. The practical effectiveness of these methods is confirmed through performance testing of the numerical scheme.

In this paper, we study the finite-time dissipativity analysis of Markovian jump-delayed neural networks (MJDNNs). The goal is to establish less conservative results for extended dissipativity conditions for delayed MJDNNs. To achieve this, an appropriate Lyapunov-Krasovskii functional (LKF) with novel inequality like composite slack-matrix-based integral inequality (CSMBII). Next, the CSMBII and other sufficient conditions are employed to estimate the derivative of the constructed LKF. Using these techniques, a delay-dependent finite-time dissipativity condition is derived in terms of linear matrix inequalities (LMIs). These LMIs are used to formulate the finite dissipativity condition for the delayed MJNNs. The utility of the suggested approach is then confirmed by a number of interesting numerical examples, one of which has been confirmed by a real-world application of the benchmark problem that is associated with the designed MJDNNs. The illustrative simulation results conclusively demonstrate the superior performance and success of the developed CSMBII technique in this proposal, surpassing the limitations of existing techniques.

This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.

The issue of extended dissipative analysis for neural networks (NNs) with additive time-varying delays (ATVDs) is examined in this research. Some less conservative sufficient conditions are obtained to ensure the NNs are asymptotically stable and extended dissipative by building the agumented Lyapunov-Krasovskii functional, which is achieved by utilizing some mathematical techniques with improved integral inequalities like auxiliary function-based integral inequalities (gives a tighter upper bound). The present study aims to solve the $ H_{\infty}, L_2-L_{\infty} $, passivity and $ (Q, S, R) $-$ \gamma $-dissipativity performance in a unified framework based on the extended dissipativity concept. Following this, the condition for the solvability of the designed NNs with ATVDs is presented in the form of linear matrix inequalities. Finally, the practicality and effectiveness of this approach were demonstrated through four numerical examples.

The goal of this paper is to find the best of two sixth-order methods, namely, RK-Huta and RK–Butcher methods for solving the fuzzy hybrid systems. We state a necessary definition and theorem in terms of consistency for convergence, and finally, we compare the obtained numerical results of two different methods with analytical solution using two different numerical examples. In addition to that, we generalize the solutions obtained by RK-6 Huta and RK-6 Butcher methods (same order different stage methods) for both the problems we handled. We are proposing these two methods in order to reduce the error in accuracy and to establish these two methods are better than any other existing numerical methods. The best of two sixth-order methods are found by the error analysis study for both the problems. Also, we show whether the change in number of stages of same order methods affects the accuracy of the approximation or not.

In this article, we investigate new results of existence and uniqueness for systems of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order and along with new kinds of coupled discrete (multi-points) and fractional integral (Riemann-Liouville) boundary conditions. Our investigation is mainly based on the theorems of Schaefer, Banach, Covitz-Nadler, and nonlinear alternatives for Kakutani. The validity of the obtained results is demonstrated by numerical examples.

Fear and prey refuges are two significant topics in the ecological community because they are closely associated with the connectivity of natural resources. The effect of fear on prey populations and prey refuges (proportional to both the prey and predator) is investigated in the nonlinear-type predator-harvested Leslie–Gower model. This type of prey refuge is much more sensible and realistic than the constant prey refuge model. Because there is less research on the dynamics of this type of prey refuge, the current study has been considered to strengthen the existing literature. The number and stability properties of all positive equilibria are examined. Since the calculations for the determinant and trace of the Jacobian matrix are quite complicated at these equilibria, the stability of certain positive equilibria is evaluated using a numerical simulation process. Sotomayor’s theorem is used to derive a precise mathematical confirmation of the appearance of saddle-node bifurcation and transcritical bifurcation. Furthermore, numerical simulations are provided to visually demonstrate the dynamics of the system and the stability of the limit cycle is discussed with the help of the first Lyapunov number. We perform some sensitivity investigations on our model solutions in relation to three key model parameters: the fear impact, prey refuges, and harvesting. Our findings could facilitate some biological understanding of the interactions between predators and prey.

In this paper, we study the Hyers-Ulam stability with respect to the linear differential condition of fourth order. Specifically, we treat ${\psi}$ as an interact arrangement of the differential condition, i.e., where ${\psi} \in c^4 [{\ell}, {\mu}], {\Psi} \in [{\ell}, {\mu}]$. We demonstrate that ${\psi}^{iv} ({\varkappa}) + {\xi}_1 {\psi}{'''} ({\varkappa})+ {\xi}_2 {\psi}{''} ({\varkappa}) + {\xi}_3 {\psi}' ({\varkappa}) + {\xi}_4 {\psi}({\varkappa}) = {\Psi}({\varkappa})$ has the Hyers-Ulam stability. Two examples are provided to illustrate the usefulness of the proposed method.

Abstract In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

Abstract This paper investigates singularly perturbed parabolic partial differential equations with delay in space, and the right end plane is an integral boundary condition on a rectangular domain. A small parameter is multiplied in the higher order derivative, which gives boundary layers, and due to the delay term, one more layer occurs on the rectangle domain. A numerical method comprising the standard finite difference scheme on a rectangular piecewise uniform mesh (Shishkin mesh) of N r × N t $N_{r} \times N_{t}$ elements condensing in the boundary layers is suggested, and it is proved to be parameter-uniform. Also, the order of convergence is proved to be almost two in space variable and almost one in time variable. Numerical examples are proposed to validate the theory.

Abstract Dynamical analysis of a delayed tri-trophic food chain consisting of prey, an intermediate, and a top predator is investigated in this paper. The additive Allee effect is introduced in the prey population, and it is assumed that there is a time lag due to the gestation effect in the intermediate predator. The interference among the prey and the intermediate predator is according to Holling type II, while the interaction between the intermediate and top predators follows the Crowley–Martin functional response. The local stability and bifurcation analysis of the proposed model at the interior equilibrium point are studied. Numerical simulations are provided to ensure the mathematical results.

This paper deals with the extended dissipativity and non-fragile synchronization of delayed recurrent neural networks (RNNs) with multiple time-varying delays and sampled-data control. A suitable Lyapunov-Krasovskii Functional (LKF) is built up to prove the quadratically stable and extended dissipativity condition of delayed RNNs using Jensen inequality and limited Bessel-Legendre inequality approaches. A non-fragile sampled-data approach is applied to investigate the problem of neural networks with multiple time-varying delays, which ensures that the master system synchronizes with the slave system and is designed with respect to the solutions of Linear Matrix Inequalities (LMIs). The effectiveness of the suggested approach is established by providing suitable simulations using MATLAB LMI control toolbox. Finally, numerical examples and comparative results are provided to illustrate the adequacy of the planned control scheme.

This article examines the issue of event-triggered $L_{2}-L_\infty $ filtering for network-based neutral systems via Takagi-Sugeno (T-S) fuzzy approach. A dynamic discrete event-triggered scheme (ETS) is introduced to save the limited communication resource. Based on the T-S fuzzy model, the consider neutral type system with networked induced delays are represented as a class of T-S fuzzy system. In addition, by considering a suitable Lyapunov-Krasovskii functional (LKF) and by using the Wirtinger inequality technique, the stability conditions with respect to linear matrix inequalities (LMIs) are presented to guarantee the considered filtering systems are asymptotically stable with $L_{2}-L_{\infty }$ performance index $\gamma $ . To the end, numerical examples are given to illustrate the effectiveness of the proposed result.

Abstract In the current study, we conduct an investigation into the Hyers–Ulam stability of linear fractional differential equation using the Riemann–Liouville derivatives based on fractional Fourier transform. In addition, some new results on stability conditions with respect to delay differential equation of fractional order are obtained. We establish the Hyers–Ulam–Rassias stability results as well as examine their existence and uniqueness of solutions pertaining to nonlinear problems. We provide examples that indicate the usefulness of the results presented.

Impact of non-uniform heat source/sink and elastic deformation on entropy generation analysis for Cu − Fe3O4/ethylene glycol hybrid nanofluid flow past a stretching sheet with an inclined magnetic field, partial slip, and thermal radiations are investigated. Appropriate conversions are utilized for PDE's into ODE's. The dimensionless form is then analytically solved with the help of a hypergeometric function. Performance of significant variables on the hybrid nanofluid flow, heat distribution, skin friction coefficient, Nusselt number, and entropy generation are obtained and discussed with the help of different graphs. Some main results reported in this article reveals that the presents of hybrid nanoparticle, partial slip, inclined magnetic field, and Eckert number improve the more heat conduction in Cu − Fe3O4/ethylene glycol hybrid nanofluid and Partial slip, non-uniform heat source, and Eckert numbers are minimized the entropy production. Skin friction coefficient improves through higher values of Cupper nanoparticles and suction parameter. Heat transfer rate is more for higher values of Cupper nanoparticles, non-uniform heat source, partial slip, thermal radiation, and Eckert number.

In this article, we propose the Holling type-II predator-prey model involving cannibalism and gestation delay in predators. We study the existence of all possible equilibrium points of the proposed model. We give the condition for the local stability and Hopf bifurcation analysis for the nondelayed model. Next, we also establish the local stability and Hopf bifurcation analysis for the corresponding delayed model. Finally, we discuss how cannibalism and delay play an important role in stabilizing and destabilizing the proposed system both theoretically and numerically.

The discrete-time system has more complex and chaotic dynamical behaviors as compared to the continuous-time system. This paper extends a discrete Leslie–Gower predator-prey system with the Allee effect in the predator’s population, whose dynamics are analyzed and explored. We have determined the equilibrium points and studied their local stability properties. We find that the system undergoes flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point by choosing the Allee parameter as a bifurcation parameter. We discuss the stability and direction of both bifurcations with the help of the normal form theory and center manifold theorem. The flip bifurcation and Neimark–Sacker bifurcation are the most common routes to the chaotic orbit in the discrete system. Moreover, we utilize state feedback, pole placement, and hybrid control methods to control the chaos in the system. The work is complete with the numerical simulations to confirm the analytical findings.

The purpose of this article is to discuss the existence, uniqueness, and Ulam–Hyers stability of solutions to a coupled system of fractional differential equations with Erdélyi–Kober and Riemann–Liouville integral boundary conditions. The Banach fixed point theorem is used to prove the uniqueness of solutions, while the Leray–Schauder alternative is used to prove the existence of solutions. Furthermore, we conclude that the solution to the discussed problem is Hyers–Ulam stable. The results are illustrated with examples.

In this paper, we investigate the general solution of a new quadratic functional equation of the form ∑1≤i

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

This paper investigates the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays. By constructing a proper Lyapunov-functional we established sufficient conditions to ensure global exponential stability of the fractional order complex-valued neural networks. The stability conditions are established in terms of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the obtained results.

This study investigated the flow and heat transfer of sodium alginate-based hybrid nanofluids with a stretching/shrinking surface. The heat source/sink, Joule heating, inclined magnetic field, and thermal radiation influences are also examined in the designed model. The mixers of non-magnetic and magnetic nanoparticles are utilized, such as Cu and Fe3O4. The Casson fluid model is applied to determine the viscoplastic characteristics of sodium alginate (SA). The necessary governing SA-based hybrid nanofluid flow equations are solved analytically by hypergeometric function. SA-based hybrid nanofluid velocity, temperature, skin friction, and Nusselt number results are discussed in detail with various pertinent parameters, such as radiation, heat source/sink, inclined angle, magnetic field, Eckert number, and Casson parameters. It is noted that the dimensions of both Cu and Fe3O4 hybrid nanoparticles and Casson parameters are minimized by the momentum surface layer thickness. The magnetic field, radiation, heat source and Casson parameters serve to enhance the thermal boundary layer thickness. Finally, the current result was verified with previously published works.

Wireless sensor networks (WSNs) carry noteworthy pros over traditional communication. Though, unkind and composite environments fake great challenges in the reliability of WSN communications. It is more vital to develop a reliable unipath dynamic source routing protocol (RDSRP)l for WSN to provide better quality of service (QoS) in energy harvesting wireless sensor networks (EH-WSN). This paper proposes a dynamic source routing approach for attaining the most reliable route in EH-WSNs. Performance evaluation is carried out using NS-2 and throughput and packet delivery ratio are chosen as the metrics.