Nonlinear Time Fractional Reaction Diffusion Equation Research Articles

Comparative analysis of three memory selection methods for time integration of Fractional Reaction-Diffusion Equations

By discretising in space, a non-linear time fractional reaction-diffusion equations (TFRDEs) can be converted into a system of time-fractional differential equations (TFDEs). The full memory method (FMM) and short memory method (SMM) are well-established memory selection methods used in the time integration of TFDEs. The main drawbacks of FMM and SMM are higher computational cost and uncontrollable error respectively. The only way to increase the accuracy of SMM is by increasing short memory length which causes an increase in computational cost. Especially when we apply these two methods to integrate TFRDEs, we have to solve a large system of TFDEs. Therefore, the drawbacks of these two methods affect seriously, when these are applied to solve TFRDEs. This paper aims to investigate the accuracy and efficiency of the memory selection method, Exponentially Decreasing Random Memory Method (EDRMM), and compare it with FMM and SMM when these methods apply to integrate TFRDEs. Based on these three memory selection methods, three semi-implicit numerical schemes namely semiimplicit scheme with full memory method (SI-FMM), semi-implicit scheme with short memory method (SI-SMM), and semi-implicit scheme with exponentially decreasing random memory method (SI-EDRMM)) are proposed and the accuracy and CPU time (computational time (CT)) of these three numerical schemes are compared. To do this comparison, these three numerical schemes are applied to four TFRDEs whose exact solutions are known. Numerical experiments confirm that the accuracy and efficiency of the SI-EDRMM are better than that of SI-SMM and the efficiency of SI-EDRMM is higher than that of SI-FMM. Therefore, EDRMM is better than SMM and FMM for the integration of TFRDEs.

A fourth order accurate numerical method for non-linear time fractional reaction–diffusion equation on a bounded domain

In the present work, a high-order numerical scheme is proposed and analyzed to solve the non-linear time fractional reaction–diffusion equation (RDE) of order α∈(0,1). The numerical scheme consists of a time-stepping cubic approximation for the time fractional derivative and a compact finite difference scheme to approximate the spatial derivative. After applying these approximations to time fractional RDE, we get a non-linear system of equations. An iterative algorithm is formulated to solve the obtained nonlinear discrete system. We analyze the unique solvability of the proposed compact finite difference scheme and discuss the stability using von Neumann analysis. Further, we prove that the scheme is convergent in the Euclidean norm with the convergence order 4−α in the temporal direction and 4 in the spatial direction using matrix analysis. Finally, the numerical experimentation is performed to demonstrate the authenticity of the proposed numerical scheme.

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

AbstractIn this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.

A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

A high-order numerical scheme for solving nonlinear time fractional reaction-diffusion equations with initial singularity

We propose a high-order numerical scheme for nonlinear time fractional reaction-diffusion equations with initial singularity, where L2-1σ scheme on graded mesh is used to approximate Caputo fractional derivative and Legendre spectral method is applied to discrete spatial variable. We give the priori estimate, existence and uniqueness of numerical solution. Then the unconditional stability and convergence are proved. The rate of convergence is O(M−min⁡{rα,2}+N−m), which is obtained without extra regularity assumption on the exact solution. Numerical results are given to confirm the sharpness of error analysis.

Final value problem for nonlinear time fractional reaction–diffusion equation with discrete data

In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reaction–diffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the quasi-boundary value method (in the case of multi-dimensional). Finally, convergence rates of the regularized solutions are given and investigated numerically.

Exponentially fitted methods for solving time fractional nonlinear reaction–diffusion equation

In this article, we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction–diffusion equation with fixed order and distributed order derivatives. These techniques depend on a parameter, which will be used to annihilate the error and increase the order of accuracy. The proposed methods are proved to be unconditionally stable and convergent by Fourier analysis. Also, the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented.

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.

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.

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.

An analytic algorithm for time fractional nonlinear reaction–diffusion equation based on a new iterative method

A new analytic algorithm for highly nonlinear time fractional reaction–diffusion equations is proposed in this paper. The proposed method is an amalgamation of variational iteration method (VIM), Adomian decomposition method (ADM) and further refined by introducing a new correction functional. This new correction functional is obtained from the standard correction functional of VIM by introducing an auxiliary parameter γ and an auxiliary function H(x) in it. Further, a sequence Gn(x,t), with suitably chosen support, is also introduced in the new correction functional. The algorithm is easy to implement and only four to six iterations are sufficient for fairly accurate solutions. The algorithm is tested on Fitzhugh – Nagumo and generalized Fisher equations with nonlinearity ranging from 2 to 5.