Nonlinear Fractional Diffusion Equation Research Articles

Numerical solution of the one-dimensional fractional convection diffusion equations based on Chebyshev operational matrix.

In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent.

Symmetry classification of time-fractional diffusion equation

In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional diffusion equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional diffusion equation transforms to a partial differential equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional diffusion equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α. This study provides a new way of computing symmetries for a class of fractional differential equations.

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

We consider nonlinear parabolic equations involving fractional diffusion of the form $${\partial_t u + {(-\Delta)}^{s} \Phi(u)= 0,}$$ with $${0 < s < 1}$$ , and solve an open problem concerning the existence of solutions for very singular nonlinearities $${\Phi}$$ in power form, precisely $${\Phi'(u)=c\,u^{-(n+1)}}$$ for some $${0 < n < 1}$$ . We also include the logarithmic diffusion equation $${\partial_t u + {(-\Delta)}^{s} \log(u)= 0}$$ , which appears as the case $${n=0}$$ . We consider the Cauchy problem with nonnegative and integrable data $${u_0(x)}$$ in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The limit solutions we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, which are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as $${t\to\infty}$$ ) of the solutions with general integrable data. A new comparison principle is introduced.

Non-existence and instantaneous extinction of solutions for singular nonlinear fractional diffusion equations

We show non-existence of solutions of the Cauchy problem in $$\mathbb {R}^N$$ for the nonlinear parabolic equation involving fractional diffusion $$\partial _t u + {{\mathrm{(-\Delta )}}}^{s}\phi (u)= 0,$$ with $$0<s<1$$ and very singular nonlinearities $$\phi $$ . It is natural to consider nonnegative data and solutions. More precisely, we prove that when $$\phi (u)=-1/u^n$$ with $$n>0$$ , or $$\phi (u) = \log u$$ , and we take nonnegative $$L^1$$ initial data, there is no solution of the problem in any dimension $$N\ge 2$$ . In one space dimension the situation is not so radical, and we find the optimal range of non-existence when $$N=1$$ in terms of s and n. As a complement, non-existence is then proved for more general nonlinearities $$\phi $$ , and it is also extended to the related elliptic problem of nonlinear nonlocal type: $$u + {{\mathrm{(-\Delta )}}}^{s}\phi (u) = f$$ with the same type of nonlinearity $$\phi $$ .

On a class of non-linear delay distributed order fractional diffusion equations

In this paper, we consider a numerical scheme for a class of non-linear time delay fractional diffusion equations with distributed order in time. This study covers the unique solvability, convergence and stability of the resulted numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ+(Δα)4+h4) in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

Симметрийная редукция и инвариантные решения нелинейного дробно-дифференциального уравнения аномальной диффузии с источником

Симметрийная редукция и инвариантные решения нелинейного дробно-дифференциального уравнения аномальной диффузии с источником

Fractional nonlinear degenerate diffusion equations on bounded domains part I. Existence, uniqueness and upper bounds

We investigate quantitative properties of nonnegative solutions u(t,x)≥0 to the nonlinear fractional diffusion equation, ∂tu+LF(u)=0 posed in a bounded domain, x∈Ω⊂RN, with appropriate homogeneous Dirichlet boundary conditions. As L we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian (−Δ)s, 0<s<1, in a bounded domain with zero Dirichlet boundary conditions, but it also includes many other examples since our theory only needs some basic properties that are typical of “linear heat semigroups”. The nonlinearity F is assumed to be increasing and is allowed to be degenerate, the prototype is the power case F(u)=|u|m−1u, with m>1.In this paper we propose a suitable class of solutions of the equation, and cover the basic theory: we prove existence, uniqueness of such solutions, and we establish upper bounds of two forms (absolute bounds and smoothing effects), as well as weighted-L1 estimates. The class of solutions is very well suited for that work. The standard Laplacian case s=1 is included and the linear case m=1 can be recovered in the limit.In a companion paper (Bonforte and Vázquez, in preparation), we will complete the study with more advanced estimates, like the upper and lower boundary behaviour and Harnack inequalities, for which the results of this paper are needed.

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.

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.

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.

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.

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.

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.

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.