Time Fractional Reaction Diffusion Equation Research Articles

Convergence and stability of compact finite difference method for nonlinear time fractional reaction–diffusion equations with delay

This paper is concerned with numerical solutions of nonlinear time fractional reaction–diffusion equations with time delay. A linearized compact finite difference scheme is proposed to solve the equations. In terms of a new developed fractional Gronwall type inequality, convergence and stability of the proposed scheme are obtained. Numerical experiments are given to illustrate the theoretical results.

Gauss-Lobatto-Legendre-Birkhoff pseudospectral scheme for the time fractional reaction–diffusion equation with Neumann boundary conditions

ABSTRACTIn this paper, an efficient numerical scheme for the time fractional reaction–diffusion equation with Neumann boundary conditions is proposed combining finite-difference method in time and Gauss-Lobatto-Legendre-Birkhoff (GLLB) pseudospectral method in space. GLLB quadrature formula involves first-order derivative values at endpoints, which allows a natural and exact imposition of Neumann boundary conditions. It is proved that the scheme is unconditionally stable and convergent with order , where τ, N and m stand for the time step, polynomial degree and spatial regularity of the exact solution. Numerical experiment is carried out to support theoretical analysis.

WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation

In this paper, a new numerical approximation is discussed for the two-dimensional distributed-order time fractional reaction–diffusion equation. Combining with the idea of weighted and shifted Grunwald difference (WSGD) approximation (Tian et al. in Math Comput 84:1703–1727, 2015; Wang and Vong in J Comput Phys 277:1–15, 2014) in time, we establish orthogonal spline collocation (OSC) method in space. A detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order $$\mathscr {O}(\tau ^2+\Delta \alpha ^2+h^{r+1})$$ , where $$\tau , \Delta \alpha , h$$ and r are, respectively the time step size, step size in distributed-order variable, space step size, and polynomial degree of space. Interestingly, we prove that the proposed WSGD-OSC scheme converges with the second-order in time, where OSC schemes proposed previously (Fairweather et al. in J Sci Comput 65:1217–1239, 2015; Yang et al. in J Comput Phys 256:824–837, 2014) can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Some numerical results are also given to confirm our theoretical prediction.

A class of efficient difference method for time fractional reaction–diffusion equation

The fractional reaction–diffusion equation has profound physical background and theoretical connotation, and its numerical methods are of great scientific significance and practical value. In this paper, a class of explicit–implicit (E–I) difference method and implicit–explicit (I–E) difference method are proposed for one-dimensional time fractional reaction–diffusion equation. The methods are constructed by combining classical explicit scheme and classical implicit scheme. The existence, uniqueness and convergence of solutions are given for E–I and I–E schemes. Theoretical analysis and numerical experiments show that the E–I and I–E schemes are unconditionally stable, with second-order spatial accuracy and \(2-\alpha \) order temporal accuracy. Compared with the classical implicit difference method with the similar accuracy, the computation time of the presented methods is improved by nearly 41\(\%\). It is shown that the E–I difference method and I–E difference method are effective for solving the time fractional reaction–diffusion equation.

Boundary Feedback Stabilization for an Unstable Time Fractional Reaction Diffusion Equation

In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove the existence and uniqueness and the Mittag--Leffler stability for the closed-loop systems. For both cases, the observers and the observer-based output feedback are designed to stabilize the systems.

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.

Numerical solution of the time fractional reaction–diffusion equation with a moving boundary

A fractional reaction–diffusion model with a moving boundary is presented in this paper. An efficient numerical method is constructed to solve this moving boundary problem. Our method makes use of a finite difference approximation for the temporal discretization, and spectral approximation for the spatial discretization. The stability and convergence of the method is studied, and the errors of both the semi-discrete and fully-discrete schemes are derived. Numerical examples, motivated by problems from developmental biology, show a good agreement with the theoretical analysis and illustrate the efficiency of our method.

Study on fractional non-autonomous evolution equations with delay

In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ(t,s), φ(t,η) and U(t), which are generated by the operator −A(t) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k-set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator −A(t) generates an analytic semigroup for every t>0. The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

Analytic solution of generalized space time fractional reaction diffusion equation

Analytic solution of generalized space time fractional reaction diffusion equation

Analytic solution of generalized space time fractional reaction diffusion equation

Analytic solution of generalized space time fractional reaction diffusion equation

Solutions of the time fractional reaction–diffusion equations with residual power series method

In this article, the residual power series method for solving nonlinear time fractional reaction–diffusion equations is introduced. Residual power series algorithm gets Maclaurin expansion of the solution. The algorithm is tested on Fitzhugh–Nagumo and generalized Fisher equations with nonlinearity ranging. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components using Mathematica software package. Reliability of the method is given by graphical consequences, and series solutions are used to illustrate the solution. The found consequences show that the method is a powerful and efficient method in determination of solution of the time fractional reaction–diffusion equations.

Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain

The numerical approximation of the distributed order time fractional reaction-diffusion equation on a semi-infinite spatial domain is discussed in this paper. A fully discrete scheme based on finite difference method in time and spectral approximation using Laguerre functions in space is proposed. The scheme is unconditionally stable and convergent with order O ( ? 2 + Δ α 2 + N ( 1 - m ) / 2 ) , where ?, Δα, N, and m are the time-step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. A pseudospectral scheme is also proposed and analyzed. Some numerical examples are presented to demonstrate the efficiency of the proposed scheme.

Exact solution of certain time fractional nonlinear partial differential equations

Given a time fractional nonlinear partial differential equation, we show how to derive its exact solution using invariant subspace method. This has been illustrated through time fractional diffusion convection equation, time fractional nonlinear dispersive Boussinesq equation, time fractional reaction diffusion equation of second order, time fractional thin-film equation, and time fractional quadratic wave equation. Also, we explicitly shown that time fractional nonlinear partial differential equations admit more than one invariant subspaces which in turn helps to derive more than one exact solution.

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.

A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

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.

Fractional Reaction-Diffusion Equations for Modelling Complex Biological Patterns

We consider a system of nonlinear time-fractional reaction-diffusion equations (TFRDE) on a finite spatial domain x ∈ [0, L], and time t ∈ [0, T]. The system of standard reaction-diffusion equations with Neumann boundary conditions are generalized by replacing the first-order time derivatives with Caputo time-fractional derivatives of order α ∈ (0, 1). We solve the TFRDE numerically using Grünwald-Letnikov derivative approximation for time-fractional derivative and centred difference approximation for spatial derivative. We discuss the numerical results and propose the applications of TFRDE for the modelling of complex patterns in biological systems.

An $$H^1$$ H 1 -Galerkin mixed finite element method for time fractional reaction–diffusion equation

In this article, an \(H^1\)-Galerkin mixed finite element (MFE) method for solving time fractional reaction–diffusion equation is presented. The optimal time convergence order \(O(\varDelta t^{2-\alpha })\) and the optimal spatial rate of convergence in \(H^1\) and \(L^2\)-norms for variable \(u\) and its gradient \(\sigma \) are derived. Moreover, some numerical results are shown to support our theoretical analysis.

A Domain Decomposition Method for Time Fractional Reaction-Diffusion Equation

The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N 2 M) compared with O(NM) for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, parallel computations. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method is proposed. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step. Numerical experiments are used to verify the efficiency of the obtained algorithm.