Fractional Reaction-diffusion Equations Research Articles

Blow-up solutions to the Cauchy problem of a fractional reaction-diffusion system

In this paper, we study the blow-up property of positive mild solutions to the Cauchy problem of a system of fractional reaction-diffusion equations. For the fundamental solution \(P(t,x)\) of the fractional heat operator \(\partial_{t}+(-\triangle)^{\frac{\beta}{2}}\) defined on the whole space \(R^{N}\), due to the properties of \(P(t,x)\) established by H Yosida and some estimates of \(P(t,x)\) developed by L Caffarelli and A Figalli, we first use an iteration method to establish the estimates of lower bounds of positive mild solutions; then we obtain the unboundedness of solutions for large time. Finally we give a sufficient condition that the positive mild solution to a fractional reaction-diffusion system blows up in finite time.

Computational Solutions of Distributed Order Reaction-Diffusion Systems Associated with Riemann-Liouville Derivatives

This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampé de Fériet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given.

Computational solutions of unified fractional reaction–diffusion equations with composite fractional time derivative

This paper deals with the investigation of the computational solutions of an unified fractional reaction–diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz–Feller fractional derivative and adding the function ϕ(x,t) which is a nonlinear function governing reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space–time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz–Feller space fractional derivatives are also discussed.

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.

Exact solutions of some fractional differential equations arising in mathematical biology

In the last decades Exp-function method has been used for solving fractional differential equations. In this paper, we obtain exact solutions of fractional generalized reaction Duffing model and nonlinear fractional diffusion–reaction equation. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation.

An analytic algorithm for the space-time fractional reaction-diffusion equation

In this paper, we solve the space-time fractional reaction-diffusion equation by the fractional homotopy analysis method. Solutions of different examples of the reaction term will be computed and investigated. The approximation solutions of the studied models will be put in the form of convergent series to be easily computed and simulated. Comparison with the approximation solution of the classical case of the studied modeled with their approximation errors will also be studied.

Bäcklund transformation of fractional Riccati equation and its applications to the space–time FDEs

In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.

Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative

This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated.

Distributed order reaction-diffusion systems associated with Caputo derivatives

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. [“The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus Appl. Anal. 4, 153–202 (2001); Mainardi et al. “Fox H-functions in fractional diffusion,” J. Comput. Appl. Math. 178, 321–331 (2005)] for the fundamental solution of the space-time fractional equation, including Haubold et al. [“Solutions of reaction-diffusion equations in terms of the H-function,” Bull. Astron. Soc. India 35, 681–689 (2007)] and Saxena et al. [“Fractional reaction-diffusion equations,” Astrophys. Space Sci. 305, 289–296 (2006a)] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion, and time-fractional diffusion, see Schneider and Wyss [“Fractional diffusion and wave equations,” J. Math. Phys. 30, 134–144 (1989)]. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables given in Appendix B of this paper. Fractional reaction-diffusion equations are of specific interest in physics for non-Gaussian, non-Markovian, and non-Fickian phenomena.

Ergodicity of the stochastic fractional reaction–diffusion equation

A stochastic reaction–diffusion equation with fractional dissipation is considered. The smaller the exponent of the equation is, the weaker the dissipation of the equation is. The equation is discussed in detail when the exponent changes. The aim is to prove the well-posedness, existence and uniqueness of an invariant measure as well as strong law of large numbers and convergence to equilibrium. Without the analytic property of the semigroup, some methods are used to overcome the difficulties to get the energy estimates. The results in this paper can be applied to the classic reaction–diffusion equation with Wiener noise.

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.

Weighted pseudo almost automorphic classical solutions and optimal mild solutions for fractional differential equations and application in fractional reaction–diffusion equations

In this paper, we are concerned with a class of fractional differential equations given by $$\begin{aligned} \hbox {D}_{t}^{\alpha }x(t)=Ax(t)+f(t,x(t)). \end{aligned}$$ Our main results concern the existence, uniqueness of weighted pseudo-almost automorphic classical solutions and optimal mild solutions. Moreover, as example and applications, we study the weighted pseudo-almost automorphic classical solutions and optimal mild solutions for a fractional reaction–diffusion equation to illustrate the practical usefulness of the analytical results that we establish in the paper.

The Decay of mass for a nonlinear fractional reaction–diffusion equation

AbstractWe study the large time behavior of non‐negative solutions to the nonlinear fractional reaction–diffusion equation ∂tu = − tσ( − Δ)α ∕ 2u − h(t)up (α ∈ (0,2]) posed on and supplemented with an integrable initial condition, where σ ≥ 0, p > 1, and h : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function h, we show that the asymptotic behavior of the mass can be classified along two cases as follows: if , then there exists M ∞ ∈ (0, ∞ ) such that ; if , then . Copyright © 2014 John Wiley & Sons, Ltd.

An efficient parallel solution for Caputo fractional reaction–diffusion equation

The computational complexity of Caputo fractional reaction—diffusion equation is $$O(MN^2)$$ O ( M N 2 ) compared with $$O(MN)$$ O ( M N ) of traditional reaction—diffusion equation, where $$M$$ M , $$N$$ N are the number of time steps and grid points. A efficient parallel solution for Caputo fractional reaction—diffusion equation with explicit difference method is proposed. The parallel solution, which is implemented with MPI parallel programming model, consists of three procedures: preprocessing, parallel solver and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector vector addition and constant vector multiplication. The sum of constant vector multiplication is optimized. As to the authors' knowledge, this is the first parallel solution for Caputo fractional reaction—diffusion equation. The experimental results show that the parallel solution compares well with the analytic solution. The parallel solution on single Intel Xeon X5540 CPU runs more than three times faster than the serial solution on single X5540 CPU core, and scales quite well on a distributed memory cluster system.

A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

Solving the Caputo Fractional Reaction-Diffusion Equation on GPU

We present a parallel GPU solution of the Caputo fractional reaction-diffusion equation in one spatial dimension with explicit finite difference approximation. The parallel solution, which is implemented with CUDA programming model, consists of three procedures: preprocessing, parallel solver, and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector-vector addition, and constant vector multiplication. The most time consuming loop of vector-vector addition and constant vector multiplication is optimized and impressive performance improvement is got. The experimental results show that the GPU solution compares well with the exact solution. The optimized GPU solution on NVIDIA Quadro FX 5800 is 2.26 times faster than the optimized parallel CPU solution on multicore Intel Xeon E5540 CPU.

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.

Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations

We study the nonlinear fractional reaction-diffusion equation $∂_t u + (-\Delta)^s u = f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\mathbb{R}^N \setminus \Omega$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $\Omega$ is symmetric with respect to reflection at a hyperplane, say $\{x_1=0\}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak additional technical assumptions, we then show that any nonzero element in the $\omega$-limit set of nonnegative globally bounded solution is even in $x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$.

Application of first integral method to fractional partial differential equations

In this paper, fractional derivatives in the sense of modified Riemann-Liouville derivative and first integral method are applied for constructing exact solutions of nonlinear fractional generalized reaction duffing model and nonlinear fractional diffusion reaction equation with quadratic and cubic nonlinearity. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solutions are constructed through established first integrals.

Two reliable wavelet methods to Fitzhugh–Nagumo (FN) and fractional FN equations

Fractional reaction–diffusion equations serve as more relevant models for studying complex patterns in several fields of nonlinear sciences. In this paper, we have developed the wavelet methods to find the approximate solutions for the Fitzhugh–Nagumo (FN) and fractional FN equations. The proposed method techniques provide the solutions in rapid convergence series with computable terms. To the best of our knowledge, until now there is no rigorous wavelet solutions have been reported for the FN and fractional FN equations arising in gene propagation and model. With the help of Laplace operator and Legendre wavelets operational matrices, the FN equation is converted into an algebraic system. Finally, we have given some numerical examples to demonstrate the validity and applicability of the wavelet methods. The power of the manageable method is confirmed. Moreover, the use of the wavelet methods is found to be accurate, efficient, simple, low computation costs and computationally attractive.