Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

Two compartmental fractional derivative model with fractional derivatives of different order

This study presents a new two compartmental model that contains fractional derivatives of different order. The model is so formulated that the mass balance is preserved. In this way we give an answer on a claim that such a model is not possible. The generalization that includes nonlinear terms and fractional order dynamics between compartments is also discussed. We derived a physically correct two compartmental system (i.e., the one preserving the mass balance) by the use of the Taylor expansion formula. Moreover, we derived in the Appendix such a system preserving the mass balance without the use of the Taylor formula but by the use of mass balance for systems with equal order fractional derivatives.     

Exact travelling wave solutions of reaction-diffusion models of fractional order

Reaction-diffusion models are used in different areas of chemistry problems. Also, coupled reaction-diffusion systems describing the spatio- temporal dynamics of competition models have been widely applied in many real world problems. In this paper, we consider a coupled fractional system with diffusion and competition terms in ecology, and reaction-diffusion growth model of fractional order with Allee effect describing and analyzing the spread dynamic of a single population under different dispersal and growth rates. Finding the exact solutions of such models are very helpful in the theories and numerical studies. Exact traveling wave solutions of the above reaction-diffusion models are found by means of the $Q$-function method. Moreover, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviours.     

Synchronization of different fractional order chaotic systems using active control

Synchronization of fractional order chaotic dynamical systems is receiving increasing attention owing to its interesting applications in secure communications of analog and digital signals and cryptographic systems. In this article we utilize active control technique to synchronize different fractional order chaotic dynamical systems. Further we investigate the interrelationship between the (fractional) order and synchronization in different chaotic dynamical systems. It is observed that synchronization is faster as the order tends to one.     

LDG method for reaction-diffusion dynamical systems with time delay

In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction-diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.     

Synchronization of nonlinear fractional order systems

This paper deals with the master-slave synchronization scheme for partially known nonlinear fractional order systems, where the unknown dynamics is considered as the master system and we propose the slave system structure which estimates the unknown state variables. For solving this problem we introduce a Fractional Algebraic Observability (FAO) property which is used as a main tool in the design of the master system. As numerical examples we consider a fractional order Rössler hyperchaotic system and a fractional order Lorenz chaotic system and by means of some simulations we show the effectiveness of the suggested approach.     

Few fractional order derivatives and their computations

This work presents an introductory development of fractional order derivatives and their computations. Historical development of fractional calculus is discussed. This paper presents how to obtain computational results of fractional order derivatives for some elementary functions. Computational results are illustrated in tabular and graphical forms. A few observations regarding integer and fractional order derivatives are mentioned in the conclusion.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

Analysis of fractional order differential coupled systems

In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B‐vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Lie symmetry analysis to Fisher's equation with time fractional order

The aim of this letter is to apply the Lie group analysis method to the Fisher''s equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher''s(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher''s is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.     

Solvability of fractional order differential systems with general nonlocal conditions

In this paper, fractional order nonlinear differential systems with general nonlocal conditions are investigated. The Lipschitz condition, linear and nonlinear growth conditions on the nonlinear terms are divided into two parts on two intervals; fixed point methods, the techniques on matrix and vector norms are used. Existence results for the solutions are derived under the weaken conditions.     

Approximate controllability of fractional order semilinear systems with bounded delay

In this paper, sufficient conditions are established for the approximate controllability of a class of semilinear delay control systems of fractional order. The existence and uniqueness of mild solution of the system is also proved. The results are obtained by using contraction principle and the Schauder fixed point theorem. Some examples are given to illustrate the theory.     

Finite difference reaction-diffusion systems with coupled boundary conditions and time delays

This paper is concerned with finite difference solutions of a coupled system of reaction-diffusion equations with nonlinear boundary conditions and time delays. The system is coupled through the reaction functions as well as the boundary conditions, and the time delays may appear in both the reaction functions and the boundary functions. The reaction-diffusion system is discretized by the finite difference method, and the investigation is devoted to the finite difference equations for both the time-dependent problem and its corresponding steady-state problem. This investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions, monotone convergence of the time-dependent solution to a maximal or a minimal steady-state solution for quasimonotone functions, and local and global attractors of the time-dependent system, including the convergence of the time-dependent solution to a unique steady-state solution. Also discussed are some computational algorithms for numerical solutions of the steady-state problem when the reaction function and the boundary function are quasimonotone. All the results for the coupled reaction-diffusion equations are directly applicable to systems of parabolic-ordinary equations and to reaction-diffusion systems without time delays.     

Convergence of a splitting method of high order for reaction-diffusion systems

In this article, we prove the convergence of a splitting scheme of high order for a reaction-diffusion system of the form where is an matrix whose spectrum is included in 0 \}$">. This scheme is obtained by applying a Richardson extrapolation to a Strang formula.

     

Triple pendulum model involving fractional derivatives with different kernels

The aim of this work is to study the non-local dynamic behavior of triple pendulum-type systems. We use the Euler-Lagrange and the Hamiltonian formalisms to obtain the dynamic models, based on the Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivative definitions. In these representations, an auxiliary parameter σ is introduced, to define the equations in a fractal temporal geometry, which provides an entire new family of solutions for the dynamic behavior of the pendulum-type systems. The phase diagrams allow to visualize the effect of considering the fractional order approach, the classical behavior is recovered when the order of the fractional derivative is 1.     

Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays

This paper is concerned with three 3-species time-delayed Lotka-Volterra reaction-diffusion systems and their corresponding ordinary differential systems without diffusion. The time delays may be discrete or continuous, and the boundary conditions for the reaction-diffusion systems are of Neumann type. The goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems. These conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays. The result of global asymptotic stability implies that each of the three model systems coexists, is permanent, and the trivial and all semitrivial solutions are unstable. Our approach to the problem is based on the method of upper and lower solutions for a more general reaction-diffusion system which gives a common framework for the 3-species model problems. Some global stability results for the 2-species competition and prey-predator reaction-diffusion systems are included in the discussion.