Time Fractional Reaction Diffusion Research Articles

Exact solutions of generalized nonlinear time-fractional reaction–diffusion equations with time delay

In this paper, we propose the invariant subspace approach to find exact solutions of time-fractional partial differential equations (PDEs) with time delay. An algorithmic approach of finding invariant subspaces for the generalized non-linear time-fractional reaction-diffusion equations with time delay is presented. We show that the fractional reaction-diffusion equations with time delay admit several invariant subspaces which further yields several distinct analytical solutions. We also demonstrate how to derive exact solutions for time-fractional PDEs with multiple time delays. Finally, we extend the invariant subspace method to more generalized time-fractional PDEs with non-linear terms involving time delay.

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville (RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.

A local discontinuous Galerkin method for time-fractional diffusion equation with discontinuous coefficient

In this paper, a fully discrete Local discontinuous Galerkin (LDG) method is discussed for a time-fractional reaction diffusion initial-boundary value problem with discontinuous diffusive coefficient. In this method the well-known L2-1σ scheme on graded meshes is presented to deal with the weak singularity at initial time t=0, while in the spatial direction a LDG method on a uniform mesh is presented to handle the discontinuous coefficient. By the discrete fractional Gronwall inequality, the L2-norm stability and consistency estimate results are derived for the proposed fully discrete LDG method. Numerical experiments are presented to verify the sharpness of the error analysis.

Mathematical Analysis and Numerical Simulation of Chaotic Noninteger Order Differential Systems with Riemann-Liouville Derivative

This article deals with the design, analysis, and implementation of a robust numerical scheme when applied to time-fractional reaction-diffusion system. Stability analysis and numerical treatment of chaotic fractional differential system in Riemann-Liouville sense are considered in this article. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise. Numerical results presented agree with the theoretical findings.

Trajectory planning for semilinear time-fractional reaction-diffusion systems under Robin boundary conditions

This paper describes how to design flatness-based distributed feedforward controllers for the solution of the trajectory planning problem of semilinear time-fractional reaction-diffusion systems (TFRDSs), where the first-order time derivative of conventional reaction-diffusion system is extended to a Caputo fractional derivative of order α ϵ (0,1]. To this end, an implicit system variable and control input parametrization is determined based on the spectral property of system operator with respect to a basic output and its fractional-order derivatives. The convergence of the parametrizations is guaranteed by restricting the basic output to some certain Gevrey classes. With these, we propose two approaches on solving the trajectory planning problem within a prescribed finite time interval. A simulation example is finally included to illustrate our results.

A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method

A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method is presented in this paper. The proposed scheme is a combination of Crank–Nicolson method for the Caputo time derivative and exponential B-spline method for space derivative. The unconditional stability and convergence of the proposed scheme are presented. Several numerical examples are presented to illustrate the feasibility and efficiency of the proposed scheme.

Mathematical modelling of pattern formation in activator–inhibitor reaction–diffusion systems with anomalous diffusion

Auto-wave solutions in nonlinear time-fractional reaction–diffusion systems are investigated. It is shown that stability of steady-state solutions and their subsequent evolution are mainly determined by the eigenvalue spectrum of a linearized system and level of anomalous diffusion (orders of fractional derivatives). The results of linear stability analysis are confirmed by computer simulations. To illustrate the influence of anomalous diffusion on stability properties and possible dynamics in fractional reaction–diffusion systems, we generalized two classical activator–inhibitor nonlinear models: FitzHugh–Nagumo and Brusselator. Based on them a common picture of typical nonlinear solutions in nonlinear incommensurate time-fractional activator–inhibitor systems is presented.

A direct discontinuous Galerkin method for time-fractional diffusion equation with discontinuous diffusive coefficient

In this work, a time-fractional reaction diffusion initial-boundary value problem with discontinuous diffusive coefficient is considered. A fully discrete direct discontinuous Galerkin (DDG) method is presented to solve it. In this method, the well-known L1 scheme with graded mesh is presented to deal with the weak singularity at initial time t = 0, while in spatial a DDG method with uniform mesh is presented to handle the discontinuous diffusive coefficient. Then norm stability and consistency estimate results are derived. Numerical experiments are presented to confirm the sharpness of the error analysis.

Numerical simulation of fractional-order reaction–diffusion equations with the Riesz and Caputo derivatives

The present paper deals with the numerical solution of space–time-fractional reaction–diffusion problems used to model some complex phenomena that are governed by dynamic of anomalous diffusion. The time- and space-fractional reaction–diffusion equation is modeled by replacing the first-order derivative in time and the second-order derivative in space, respectively, with the Caputo and Riesz operators. We propose an adaptable numerical scheme for the approximation of the derivatives. Accuracy of the method is justified by reporting both the maximum error and relative error in 2D with analytical solutions given. The effectiveness and applicability of the proposed methods are tested on two of practical problems that are of current and recurring interests in one and two dimensions to cover pitfalls that may arise.

A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

On the complete synchronization of a time-fractional reaction–diffusion system with the Newton–Leipnik nonlinearity

ABSTRACT In this paper, we consider a time-fractional reaction–diffusion system with the same nonlinearities of the Newton–Leipnik chaotic system. Through analytical tools and numerical results, we derive sufficient conditions for the asymptotic stability of the proposed model and show the existence of chaos. We also propose a nonlinear synchronization controller for a pair of systems and establish the local and global asymptotic convergence of the trajectories by means of fractional stability theory and the Lyapunov method.

Shifted fractional Jacobi spectral algorithm for solving distributed order time-fractional reaction–diffusion equations

This paper proposes a new method for solving distributed order time-fractional reaction–diffusion equations (DO-TFRDEs). Extended versions of the shifted Jacobi–Gauss–Lobatto and shifted fractional order Jacobi–Gauss–Radau collocation methods are developed for reducing the DO-TFRDEs to systems of algebraic equations and computing their approximate solutions. The applicability and accuracy of the method is illustrated through numerical examples.

Numerical approach to the time-fractional reaction-diffusion equation

The numerical solution to the time-fractional reaction-diffusion equation with boundary conditions is considered in this paper. By difference, the problem is transformed to solve a linear system whose coefficient matrices are Toeplitz-like, and the solution can be constructed directly. Numerical results are reported to show the feasibility of the proposed method.

Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives

In this paper, we numerically study a nonlinear time-fractional reaction-diffusion equation involving the Caputo and Atangana–Baleanu fractional derivatives of order α ∈ (0, 1). A novel algorithm known as the Laplace Adams–Bashforth method is formulated for the approximation of these derivatives. In the simulation framework, a tri-tropic food chain system is considered in which the classical time-derivatives are replaced with non-integer order derivatives. Mathematical analysis of the main system is examined for both stability and Hopf-bifurcations to occur. Numerical simulation results show the existence of chaotic behaviours and spatiotemporal oscillations as well as the emergence of some Turing patterns (such as, spots and stripes) in two-dimensional space.

A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition

A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω×[0,T], where Ω=(0,l)⊂R. The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t=0. Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L2(Ω) and H1(Ω) norms at each discrete time level tn by means of a non-trivial projection of the unknown solution into the finite element space. The L2(Ω) bound is optimal for all tn; the H1(Ω) bound is optimal for tn not close to t=0. An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp.

Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point

Abstract In this article, we study complex spatio-temporal solutions in nonlinear time-fractional reaction-diffusion systems. The main attention is paid to nonlinear dynamics near a bifurcation point. Despite the fact that the homogeneous state is stable at the parameters lower than bifurcation ones, a variety of complex solutions can also form in the subcritical domain. As an example, we consider a generalized fractional FitzHugh-Nagumo model. Depending on the given standard bifurcation parameters and the order of fractional derivative, the new types of steady auto-wave solutions in such systems have revealed. By computer simulation, it is shown that fractional reaction-diffusion possess much more complex nonlinear dynamics than their integer counterparts even at a subcritical bifurcation.

Analytical approach for the fractional differential equations by using the extended tanh method

In this study, we consider analytical solutions of space–time fractional derivative foam drainage equation, the nonlinear Korteweg–de Vries equation with time and space-fractional derivatives and time-fractional reaction–diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann–Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.

A Fully Discrete LDG Method for the Distributed-Order Time-Fractional Reaction–Diffusion Equation

In this paper, the numerical approximation of the distributed-order time-fractional reaction–diffusion equation is proposed and analyzed. Based on the finite difference method in time and local discontinuous Galerkin method in space, we develop a fully discrete scheme and prove that the scheme is unconditionally stable and convergent with order $$O\left( h^{k+\frac{1}{2}}+(\Delta t)^2+\Delta \alpha ^4\right) $$ , where $$h,k,\Delta t$$ and $$\Delta \alpha $$ are the space-step size, piecewise polynomial degree, time-step size, step size in distributed-order variable, respectively. Numerical examples are presented to show the effectiveness and the accuracy of the numerical scheme.

Fast Adomian decomposition method for the Cauchy problem of the time-fractional reaction diffusion equation

The fractional reaction diffusion equation is one of the popularly used fractional partial differential equations in recent years. The fast Adomian decomposition method is used to obtain the solution of the Cauchy problem. Also, the analytical scheme is extended to the fractional one where the Taylor series is employed. In comparison with the classical Adomian decomposition method, the ratio of the convergence is increased. The method is more reliable for the fractional partial differential equations.

Numerical approximation of distributed order reaction–diffusion equations

In this paper an implicit scheme for the numerical approximation of the distributed order time-fractional reaction–diffusion equation with a nonlinear source term is presented. The stability and the convergence order of the numerical scheme are analysed and illustrated through some numerical examples.