Nonlinear Fractional Diffusion Research Articles

Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications

AbstractIn this paper, we formulate and prove the weak and strong maximum principles for a general parabolic-type fractional differential operator with the Riemann–Liouville time-fractional derivative of distributed order. The proofs of the maximum principles are based on an estimate of the Riemann–Liouville fractional derivative at its maximum point that was recently derived by the authors. Some a priori norm estimates for solutions to initial-boundary value problems for linear and nonlinear fractional diffusion equations of distributed order and uniqueness results for these problems are presented.

Application of Sumudu Transform in Generalized Fractional Reaction–Diffusion Equation

In this paper we investigate the solution of a generalized nonlinear fractional reaction diffusion equation by the application of Sumudu Transform.

A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains

We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$ to the nonlinear fractional diffusion equation, $${\partial_t u + \mathcal{L} (u^m)=0}$$ , posed in a bounded domain, $${x\in\Omega\subset \mathbb{R}^N}$$ , with m > 1 for t > 0. As $${\mathcal{L}}$$ we use one of the most common definitions of the fractional Laplacian $${(-\Delta)^s}$$ , 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. In addition, we obtain similar estimates for fractional semilinear elliptic equations. Either the standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.

Lie group analysis method for two classes of fractional partial differential equations

In this paper we deal with two classes of fractional partial differential equation: n order linear fractional partial differential equation and nonlinear fractional reaction diffusion convection equation, by using the Lie group analysis method. The infinitesimal generators general formula of n order linear fractional partial differential equation is obtained. For nonlinear fractional reaction diffusion convection equation, the properties of their infinitesimal generators are considered. The four special cases are exhaustively investigated respectively. At the same time some examples of the corresponding case are also given. So it is very convenient to solve the infinitesimal generator of some fractional partial differential equation.

Maximum principle for the multi-term time-fractional diffusion equations with the Riemann–Liouville fractional derivatives

In this paper, the initial-boundary-value problems for linear and non-linear multi-term fractional diffusion equations with the Riemann–Liouville time-fractional derivatives are considered. To guarantee the uniqueness of solutions, we employ the weak and the strong maximum principles for the equations under consideration that are formulated and proved in this paper for the first time. An essential element of our proof of the maximum principles is an estimation for the value of the Riemann–Liouville fractional derivative of a function at its maximum point that is established in this paper for a wider space of functions compared to those used in our previous publications. In the linear case, the solutions to the problems under consideration are constructed in form of the Fourier series with respect to the eigenfunctions of the corresponding eigenvalue problems.

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.

On the numerical solution and convergence study for the system of nonlinear fractional diffusion equations

In this article, an implementation of an efficient numerical method for solving the system of coupled nonlinear fractional diffusion equations (NFDEs) is introduced. The proposed system has many applications, such as porous media and plasma transport. The fractional derivative is described in the Caputo sense. The method is based upon a combination between the properties of the Legendre approximations and finite difference method (FDM). The proposed method reduces NFDEs to a system of ordinary differential equations that are solved using FDM. Special attention is given to the study of the convergence analysis and deducing the upper bound of the error of the resulting approximate solution. A numerical example is given to show the validity and the accuracy of the proposed method.

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$ for $\alpha \in (0,2)$. Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on $\alpha$ in the $BV$-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits $\alpha \downarrow 0$ and $\alpha \uparrow 2$. In the limit $\alpha \uparrow 2$, $\Delta^{\alpha/2}$ converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231-251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).

Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications

Abstract In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

The Fisher-KPP Equation with Nonlinear Fractional Diffusion

We study the propagation properties of nonnegative and bounded solutions of the class of reaction-diffusion equations with nonlinear fractional diffusion: $u_{t} + (-\Delta)^s (u^m)=f(u)$. For all $0 m_c=(N-2s)_+/N $, we consider the solution of the initial-value problem with initial data having fast decay at infinity and prove that its level sets propagate exponentially fast in time, in contrast to the traveling wave behavior of the standard KPP case, which corresponds to putting $s=1$, $m=1$, and $f(u)=u(1-u)$. The proof of this fact uses as an essential ingredient the recently established decay properties of the self-similar solutions of the purely diffusive equation, $u_{t} + (-\Delta)^s u^m=0$.

Quantitative local and global a priori estimates for fractional nonlinear diffusion equations

We establish quantitative estimates for solutions u(t,x) to the fractional nonlinear diffusion equation, ∂tu+(−Δ)s(um)=0 in the whole range of exponents m>0, 0<s<1. The equation is posed in the whole space x∈Rd. We first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data. In the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions, depending only on the norm of the initial data in a certain ball. The estimates take a different form in three exponent ranges: slow diffusion, good range of fast diffusion, and very fast diffusion. Finally, we show existence and uniqueness of initial traces.

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.

Energy spreading in strongly nonlinear disordered lattices

We study the scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave packets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We show that the spreading in nonlinearly coupled linear oscillators slows down compared to a pure power law, while for nonlinear local oscillators a power law is valid in the whole studied range of parameters.

Solution of the Nonlinear Fractional Diffusion Equation with Absorbent Term and External Force Using Optimal Homotopy-Analysis Method

In this article, the optimal homotopy-analysis method (HAM) is used to obtain approximate analytic solutions of the time-fractional nonlinear diffusion equation in the presence of an external force and an absorbent term. The fractional derivatives are considered in the Caputo sense to avoid nonzero derivative of constants. Unlike usual HAM this method contains at the most three convergence control parameters which determine the fast convergence of the solution through different choices of convergence control parameters. Effects of proper choice of parameters on the convergence of the approximate series solution by minimizing the averaged residual error for different particular cases are depicted through tables and graphs

An approximate analytical solution of nonlinear fractional diffusion equation by homotopy analysis method

In the present paper, the approximate analytical solution of a nonlinear diffusion equation with fractional time derivative () and with the diffusion term as () are obtained with the help of analytical method of nonlinear problem called the Homotopy Analysis Method (HAM). By using initial value, the explicit solution of the equation for different particular cases have been derived which demonstrate the effectiveness, validity, potentiality and reliability of the method in reality. Numerical results of the fast and slow diffusion for different particular cases are presented graphically. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. Key words: Partial differential equation, nonlinear fractional diffusion equation, Brownian motion, homotopy analysis method.

NONEXISTENCE OF SOLUTIONS RESULTS FOR CERTAIN FRACTIONAL DIFFERENTIAL EQUATIONS

This paper is meant to establish sufficient conditions for the nonexistence of weak solutions to nonlinear fractional diffusion equation in space and time with nonlinear convective term. The Fujita exponent is determined.

Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion–wave equations

In this paper, the Laplace decomposition method is employed to obtain approximate analytical solutions of the linear and nonlinear fractional diffusion–wave equations. This method is a combined form of the Laplace transform method and the Adomian decomposition method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The fractional derivative described here is in the Caputo sense. Some illustrative examples are presented and the results show that the solutions obtained by using this technique have close agreement with series solutions obtained with the help of the Adomian decomposition method.

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection–diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection–diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.

Solution of the nonlinear fractional diffusion equation with absorbent term and external force

The article presents the approximate analytical solutions of general nonlinear diffusion equation with fractional time derivative in the presence of an absorbent term and a linear external force obtained with the help of powerful mathematical tool like Homotopy Perturbation Method. By using initial value, the approximate analytical solutions of the equation are derived. The fractional derivatives are described in the Caputo sense. Numerical results for different particular cases are presented graphically. The anomalous behavior of nonlinear diffusivity in the presence or absence of external force and reaction term are calculated numerically and presented graphically.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation ∂ u ( x , t ) ∂ t = B ( x , t ) x R α ( x , t ) u ( x , t ) + f ( u , x , t ) , where x R α ( x , t ) is a generalized Riesz fractional derivative of variable order α ( x , t ) ( 1 < α ( x , t ) ⩽ 2 ) and the nonlinear reaction term f ( u , x , t ) satisfies the Lipschitz condition | f ( u 1 , x , t ) - f ( u 2 , x , t ) | ⩽ L | u 1 - u 2 | . A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.