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Keywords : fractional differential equation; impulsive fractional differential equation; impulse; Caputo-Katugampola derivative. Finally,rn a brief conclusion is included. Results obtained in this paper have extended and improved conclusions contained in other literatures. Several illustrative examples are presented.

Department of Applied Science and Engineering,Indian Institue of Technology Roorkee

Keywords : Oscillation; Neutral; Delay; Differential equation. While the previous studies considered this effect on susceptible compartment only. The theorems of asymptotically autonomous systems and the generalized Poincare-Bendixson are used to show that the endemic equilibrium is globally asymptotically stable. Numerical methods are used to solve the obtained system of differential equations and the solutions are illustrated in several examples.

The behavior of the susceptible, exposed and infected nodes in the computer network have been analyzed. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of planar differential systems of degree seven. Under some suitable conditions, the existence of three limit cycles two of them are non-algebraic while the third is algebraic is proved.

Furthermore, these limit cycles are explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our results. Keywords : Planar polynomial differential system; First integral; Periodic orbits; algebraic and non-algebraic limit cycle. Using the stochastic analysis technique and fixed point theorem, a set of sufficient conditions is obtained for the required result of approximate controllability of stochastic integrodifferential equations of Sobolev type with unbounded delay.

Finally, an example is provided to illustrate the obtained result. Keywords : Approximate Controllability; Fixed point theorem; Stochastic differential equation; Mild solution. We convert the specified nonlinear boundary value problem with Dirichlet and Neumann boundary conditions, that governs the large deflections, to an equivalent nonlinear Fredholm-Volterra integral equation.

We illustrate the obtained approximations by appropriate graphs and examine the resulting possible errors. Finally, we discuss the relationship of the deflection and the model parameters. Keywords : large deflection; boundary value problem; flexible cantilever beam; functionally graded material; Adomian decomposition method; Adomian polynomials. Keywords : Galerkin approximation procedure; Global solution; Blow up; Potential well. Keywords : Hopf bifurcation; averaging theory; cubic polynomial differential systems;.

By investigating the corresponding characteristic equation, the local stability of a positive equilibrium and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. The critical value of evaluation period is determined beyond which small amplitude oscillations of the adopter and non-adopters population occur, and this critical value goes on decreasing with the increase in carrying capacity of the non-adopters population. Basic results are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem.

Sensitivity analysis is performed for state variables at positive steady state on model parameters. It is observed that the cumulative density of external influences has a significant role in developing the maturity stage final adoption stage in the system. Numerical computations are executed to confirm the correctness of theoretical investigations. Keywords : Innovation diffusion model; Stability analysis; Sensitivity analysis; Hopf bifurcation; Center manifold theorem; Normal form theory. Two examples illustrate the results.

The explicit formula of the least squares estimators are obtained and the estimation error is given. The simulation is made to verify the effectiveness of the estimators. Then we consider the initial value problem IVP for multi-term fractional differential equation. By introducing new unknown functions, we rewrite the IVP for multi-term fractional differential equation into the IVP for a fractional differential equation system. Thus the solution can be given in terms of matrix Mittag-Leffler functions. Keywords : fractional calculus; fractional derivative; Mittag-Leffler function; fractional differential equation.

It is assumed that in the reserved zone only prey species can access and predation is strictly prohibited, whereas in the free zone both the species can cohabit and naturally predation is allowed. The migration rates of the prey species from reserved zone to unreserved zone and vice-versa both depends on predator's availability and accordingly suitable functions has been incorporated in the model system. The local and global stability analysis of the model system have been performed in a systematic manner and system persistence criterion has been established.

The role of prey migration rate from reserved zone to unreserved zone has been investigated and it is found that Hopf bifurcation occurs when the prey migration rate from reserved zone to unreserved zone crosses a certain threshold value. It is also found that the prey migration rate has stabilizing effect on the dynamics of the system and has significant effect on the coexistence of all the species.

Finally numerical simulation has been carried out to support our analytical findings. Keywords : Prey-predator model; Reserved zone; Stability and persistence; Hopf bifurcation; Limit cycle. Motivated by recently introduced fractional operators with non-singular kernels, in this paper a comparison of the solution of linearized fractional Boussinesq equation has been made for the fractional operators Caputo with singular kernel and Caputo-Fabrizio with non-singular kernel. Linearized Boussinesq equation is derived by assuming that the average thickness of saturated layer of an aquifer is constant.

We show that due to the delayed logistic growth of the prey, it is impossible for the species to become extinct through predation. Conditions for existence and local stability of the co-existence equilibrium are derived in terms of system parameters. Using techniques of centre manifold reduction and the normal form theory, we establish the direction of Hopf bifurcation of the co-existence equilibrium, as well as the stability of the bifurcating period solution.

Numerical bifurcation analysis and simulations are performed to illustrate regions of stability of the co-existence equilibrium, to investigate how the amplitude and the period of bifurcating periodic solutions depend on parameters, and to demonstrate different types of dynamics of the system. Keywords : Stability; discrete and distributed delay; predator-prey model; Hopf bifurcation; periodic solutions.

We establish some new oscillation criteria of the solutions of the differential system by using the generalized Riccati transformation and the integral averaging method. The obtained results are illustrated by various examples. Keywords : Oscillation; Delay; Partial differential system; Conformable fractional derivative.

This model has a global positive solution.

Delay Differential Evolutions Subjected to Nonlocal Initial Conditions

Firstly, we establish sufficient conditions for extinction and persistence in the mean of a disease. Then, we prove the global stability of the system under a suitable condition of perturbation intensity. In the case of the non-autonomous system, we show that there exists at least one positive periodic solution. Finally, some numerical examples are introduced to show the validity of our results.

Keywords : Stochastic SIRS model; vertical transmission; global stability; extinction; persistence; periodic solution. Using the Mountain-Pass Theorem, we obtainrnthe existence of positive homoclinic solution in both cases sub-quadratic andrnsuper-quadratic. Cheap controls for disturbances compensation in hyperbolic delayed systems.

With a convenient choice of input operator control and through the observation output , we show how to remedy the effect of any disturbance f on the considered system. We give the main properties and characterizations of the concept according to the delay. Then, under the appropriate hypothesis, we prove howto find the optimal control ensures the compensation of a disturbance using the corresponding observation only.

The usual case of actuators and sensors is examined. An application and numerical results for a one-dimensional wave equation with delay are also presented. Three criteria of these equations are obtained for oscillation. And examples are given to show the meanings of the theorems. Keywords : Chaos control; financial system; stability; Hopf bifurcation; fractional order; delay. According to the definition of generalized fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations.

The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation.

Keywords : Residual power series method; Time-fractional Fornberg-Whitham equation; Caputo derivative. The operational matrices of integration, differentiation, production, and delay are derived and utilized to reduce the time-delay dynamical system to a set of algebraic equations. The method is easy to implement.

SIAM Journal on Control and Optimization

The illustrative examples with time-invariant and time-varying coefficients demonstrate the validity of the method. Keywords : Time-delay system; Hermite wavelet; Operational matrix; Direct method. The periodic quasi-wavelets PQWs constructed on [0,2pi] are utilized as a basis of the iteration method. Using the Banach fixed point theorem, we obtain some results concerning the error analysis. Illustrative examples are included to demonstrate the validity and applicability of the technique. Keywords : Nonlinear Fredholm integral equation; Periodic quasi-wavelet; Complex plane; fixed point theorem; error analysis.

Boukari Brahim, Hattaf Khalid, E. In the proposed model, the first distributed delays describes the time needed for infected cells to produce new virions, and the second portrays the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, we prove that the proposed discrete model has the same dynamics as the corresponding continuous model, such as positivity, boundedness and global behaviors of solutions with no restriction on the time step size.

Moreover, numerical simulations are given to illustrate and confirm our main analytical results. Keywords : Viral infection; distributed delay; difference equation; global stability. By constructing a new Lyapunov-Krasovskii functional, a novel synchronization criterion is presented in terms of matrix inequalities based on Chen's integral inequalities and reciprocal convex technique.

These established conditions are heavily dependent on the bounds of both time-delay and its derivative. Through employing Matlab Toolbox and adjusting some matrix parameters in the derived results, the design and applications of the generalized networks can be realized. The effectiveness and applicability of the proposed methods is demonstrated by a numerical example with simulations. Keywords : synchronization; matrix inequality; hybrid coupled neural networks; reciprocal convex technique. It is discussed that how the stochastic differential equations SDE could numerically be solved as matrix problems.

By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra integral equations is reduced to systems of algebraic equations with unknown Legendre coefficients. Finally, the high accuracy of approximated solutions are illustrated by several experiment. Keywords : Stochastic Volterra integral equation; Brownian motion; Approximate solution; Best approximation; Legendre polynomials; Collocation method.

By using the Riccati transformation, some inequalitiess and integral averaging technique, interval oscillation criteria of both El-Sayed type and Kong type are established. Finally, two examples are presented to illustrate the theoretical results. Keywords : Periodic solutions; Difference equations; Max-type system. The global dynamical properties like permanence and global stability of the system as well as extinction of disease are analytically and numerically studied. The impact of behavioural patterns of individuals on disease control is validated along with possible applications.

Further, Pontryagin's Maximum Principle is used to characterize optimal level of the two controls, treatment and awareness level.

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Our objective is to minimise the infected population as well as the cost of applied control. The controls at optimal level are found to achieve different levels of impact on infection. It is observed that the combined impact of treatment and awareness exhibits more effective result in disease control compared to their single application. Based on observation, the strategy regarding the implementation of awareness and treatment is suggested. We consider the cases in which the right hand side can berneither convex.

The results indicate that under a variety of conditions the variational algorithm performs at least as well as the sequential algorithm. Limitations and possible extensions, as well as operational implications of this work, are briefly discussed. Volume , Issue If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username.

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Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract The problem of data assimilation that is concerned with the complete and accurate specification of the atmospheric state based upon observations and other types of information can be approached either by variational or sequential algorithms. Citing Literature. Related Information.