Nonlinear Fractional Diffusion Equation Research Articles

Solutions for a fractional nonlinear diffusion equation with external force and absorbentterm

We devote this work to investigate the solutions of a -dimensional nonlinear fractional diffusion equation which emerges from the continuityequation by considering a nonlinear fractional generalization of Darcy law andincorporating an absorbent term. The solutions obtained show a nonusual spreading of thedistribution and a compact or long tail behavior which can be related to the anomalousdiffusion.

Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation

In this paper, we adopt the homotopy analysis method (HAM) to obtain solutions of linear and nonlinear fractional diffusion and wave equation. The fractional derivative is described in the Caputo sense. Some illustrative examples are presented.

Solutions of fractional nonlinear diffusion equation and first passage time: Influence of initial condition and diffusion coefficient

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with diffusion coefficient separable in time and space, D ( t , x ) = D ( t ) | x | − θ , subject to absorbing boundary condition and the conventional initial condition p ( x , 0 ) = δ ( x − x 0 ) . We obtain explicit analytical expressions for the probability distribution, the first passage time distribution, the mean first passage time and the mean squared displacement, and discuss their behavior corresponding to different time dependent diffusion coefficients.

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.

Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.

Solving fractional diffusion and wave equations by modified homotopy perturbation method

This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and Odibat to obtaining solutions of linear and nonlinear fractional diffusion and wave equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method.

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W ( ± ∞ , t ) = 0 and the sharp initial condition W ( x , 0 ) = δ ( x ) .

Monotone method for initial value problem for fractional diffusion equation

Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.

Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition

Adomian decomposition method has been used to obtain solutions of linear/nonlinear fractional diffusion and wave equations. Some illustrative examples have been presented.

Nonlinear fractional diffusion equation: Exact results

The nonlinear fractional diffusion equation ∂tρ=r1−d∂rμ′{rd−1D(r,t;ρ)∂rμρν}−r1−d∂r{rd−1F(r,t)ρ}+α¯(t)ρ is studied by considering the diffusion coefficient D(r,t;ρ)=D(t)r−θργ and the external force F(r,t)=−k1(t)r+kαrα. In addition, a rich class of diffusive processes, including normal and anomalous ones, is obtained from the study present in this work.

Solutions for a fractional nonlinear diffusion equation: Spatial time dependent diffusion coefficient and external forces

We analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our investigation by considering the linear case and the nonlinear case afterward. The linear case is discussed taking fractional time and spatial derivatives into account in a unified approach. We also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by D(x,t)=D(t)|x|−θ. For the nonlinear case, we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation.

Anisotropic fractional diffusion equation

We analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence, the presence of external forces and time fractional derivatives. We obtain new exact classes of solutions for a linear anisotropic fractional diffusion equation and investigate the time scaling behavior and an asymptotic solution for a nonlinear anisotropic fractional diffusion equation. We connect the asymptotic solution obtained with the distribution that emerges from the nonextensive statistics to the nonlinear case. We also verify different diffusive behavior, for instance, subdiffusion and superdiffusion, in each direction.

Fractional and nonlinear diffusion equation: additional results

We investigate the solutions of a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our study by considering the linear case and the nonlinear case afterward. The linear case is analyzed taking fractional time and spatial derivatives into account. In this context, we also discuss the modifications that emerge by considering a diffusion coefficient given by D(x)∝|x| −θ . For the nonlinear case accomplishing the fractional time derivative, we discuss scaling behavior of the time and the asymptotic for the solution of the nonlinear fractional diffusion equation. In this case, the connection between the asymptotic solution found here and the nonextensive Tsallis statistics is performed.

Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption.

We analyze a nonlinear fractional diffusion equation with absorption by employing fractional spatial derivatives and obtain some more exact classes of solutions. In particular, the diffusion equation employed here extends some known diffusion equations such as the porous medium equation and the thin film equation. We also discuss some implications by considering a diffusion coefficient D(x,t)=D(t)/x/(-theta) (theta in R) and a drift force F=-k(1)(t)x+k(alpha)x/x/(alpha-1). In both situations, we relate our solutions to those obtained within the maximum entropy principle by using the Tsallis entropy.