Time Fractional Reaction Diffusion Equation Research Articles

Analysis of a physically-relevant variable-order time-fractional reaction–diffusion model with Mittag-Leffler kernel

It is known that the well-posedness of time-fractional reaction–diffusion models with Mittag-Leffler kernel usually requires non-physical constraints on the initial data. In this paper, we propose a variable-order time-fractional reaction–diffusion equation with Mittag-Leffler kernel and prove that the aforementioned constraints could be eliminated by imposing the integer limit of the variable fractional order at the initial time, which mathematically demonstrates the physically-relevance of the variable-order modifications.

An Efficient Parallel Approximate Algorithm for Solving Time Fractional Reaction-Diffusion Equations

In this paper, we construct pure alternative segment explicit-implicit (PASE-I) and implicit-explicit (PASI-E) difference algorithms for time fractional reaction-diffusion equations (FRDEs). They are a kind of difference schemes with intrinsic parallelism and based on classical explicit scheme and classical implicit scheme combined with alternating segment technology. The existence and uniqueness analysis of solutions of the parallel difference schemes are given. Both the theoretical proof and the numerical experiment show that PASE-I and PASI-E schemes are unconditionally stable and convergent with second-order spatial accuracy and 2−α order time accuracy. Compared with implicit scheme and E-I (I-E) scheme, the computational efficiency of PASE-I and PASI-E schemes is greatly improved. PASE-I and PASI-E schemes have obvious parallel computing properties, which shows that the difference schemes with intrinsic parallelism in this paper are feasible to solve the time FRDEs.

A numerical method for solving the time fractional reaction-diffusion equation with variable coefficients on the whole line

This paper focuses on numerically solving the time fractional reaction-diffusion equation on the whole line. A numerical scheme is constructed based on Hermite pseudospectral method, we use finite difference scheme in time direction while Hermite-Gauss points in space. Several numerical results for the Hermite pseudospectral scheme are provided to confirm that a second-order time accuracy and spectral accuracy in space can be obtained.

An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation

In this paper, a new method is proposed to solve the time-fractional reaction-diffusion equation with nonlinear variable coefficients. Based on the second-order weighted and shifted Grünwald difference (WSGD) method for time Riemann-Liouville fractional derivative, we formulate and analyze the extrapolated Crank-Nicolson (ECN) method in time and orthogonal spline collocation (OSC) method in space. We prove the optimal global convergence order by a novel choice of the first step solution uh1. Finally, some numerical results are given to verify the convergence rate in the theoretical analysis.

Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method

In this paper, two-grid algorithms based on expanded mixed finite element method are presented for solving two-dimensional semilinear time fractional reaction-diffusion equations. To obtain the fully discrete scheme, the classical L1 scheme is considered in the time direction, and the expanded mixed finite element method is used to approximate spatial direction. Then the error estimates and stability of fully discrete scheme are derived. To linearize the nonlinear system, the two-grid method based on Newton iteration are constructed. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment is presented further confirm the theoretical results.

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.

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.

An operational matrix method for nonlinear variable-order time fractional reaction–diffusion equation involving Mittag-Leffler kernel

This paper is concerned with an operational matrix (OM) scheme based on the shifted Chebyshev cardinal functions (SCCFs) of the second kind for numerical solution of the variable-order time fractional nonlinear reaction–diffusion equation. The fractional derivative operator is defined in the sense of Atangana–Baleanu–Caputo. Through the way, a new OM of variable-order fractional derivative is derived for the mentioned cardinal functions. More precisely, the unknown solution is expanded by the SCCFs with undetermined coefficients. Then the expansion substituted in the equation and the generated OM is utilized to extract some algebraic equations. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm the suggested approach is highly accurate to provide satisfactory 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.

Space-time finite element method for the distributed-order time fractional reaction diffusion equations

In this paper, we propose a space-time finite element method for the distributed-order time fractional reaction diffusion equations (DOTFRDEs). First, using the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature rule to discretize the distributed-order derivative and employing the finite element method both in space and time, three fully discrete finite element schemes for DOTFRDEs are developed. Second, for the obtained numerical schemes, the existence, uniqueness and stability are discussed. Third, under the hypothesis about the singular behavior of exact solution near t=0, the convergence of these numerical schemes are investigated in detail based on the graded time mesh. Forth, in order to reduce the storage requirement and computational cost of these numerical schemes, an efficient sum-of-exponentials approximation for the kernel t−α,α∈(0,1) is introduced, and the developed space-time finite element schemes are improved. At last, some numerical tests are given to verify the rationality and effectiveness of our method.

Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

Abstract In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of nonlinear variable-order (V-O) time fractional 2D reaction-diffusion equations. The V-O time fractional derivative is defined in the Caputo sense with Mittag-Leffler non-singular kernel of order α ( x , t ) ∈ ( 0 , 1 ) (known as the Atangana–Baleanu–Caputo fractional derivative). First, the V-O fractional derivative is approximated via the finite difference scheme and the theta-weighted method is utilized to derive a recursive algorithm. Then, the unknown solution of the intended problem is expanded via the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the solution of the problem is reduced to solution of a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the applicability, validity and accuracy of the presented wavelet method, some numerical test examples are solved. The achieved numerical results reveal that the established method is highly accurate in solving the introduced new V-O fractional model.

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.

Exact traveling and non-traveling wave solutions of the time fractional reaction–diffusion equation

In this study, solutions to the nonlinear time-dependent fractional reaction–diffusion equation with conformal fractional derivative is considered. In the first part of the manuscript, we reduce the fractional equation to a traditional differential equation using the fractional complex transformation. Two cases where the exact solution exists are then presented. One of these solutions is used to model experimental data showing anomalous diffusion in freestanding graphene. In the second part, we study the traveling solutions and present two cases: Canonical-like transformation method and complete discrimination system for polynomial method.

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.

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.

A Second-Order Accurate Implicit Difference Scheme for Time Fractional Reaction-Diffusion Equation with Variable Coefficients and Time Drift Term

An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the $L_2$-norm is $\mathcal{O}(\tau^2 + h^2)$ with time step $\tau$ and mesh size $h$. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.

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.

Fractionally delineate the neuroprotective function of calbindin-D28k in Parkinson’s disease

Neuron is a fundamental unit of the brain, which is specialized to transmit information throughout the body through electrical and chemical signals. Calcium ([Formula: see text]) ions are known as second messengers which play important roles in the movement of the neurotransmitter. Calbindin-[Formula: see text] is a [Formula: see text] binding protein which is involved in regulation of intracellular [Formula: see text] ions and maintains [Formula: see text] homeostasis level, it also alters the cytosolic calcium concentration ([[Formula: see text]]) in nerve cells to keep the cell alive. Parkinson’s disease (PD) is a chronic progressive neurodegenerative brain disorder of the nervous system. Several regions of the brain indicate the hallmark of the PD. The symptoms of PD are plainly linked with the degeneration and death of dopamine neurons in the substantia nigra pars compacta located in midbrain which is accompanied by depletion in calbindin-[Formula: see text]. In the present paper, the neuroprotective role of calbindin-[Formula: see text] in the cytoplasmic [[Formula: see text]] distribution is studied. The elicitation in [[Formula: see text]] is due to the presence of low amount of calbindin-[Formula: see text] which can be portrayed and is a hallmark of PD. A one-dimensional space time fractional reaction diffusion equation is designed by keeping in mind the physiological condition taking place inside Parkinson’s brain. Computational results are performed in MATLAB.

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.