Related papers: The explicit formula for solution of anomalous dif…
Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose…
Inverse problems for a diffusion equation containing a generalized fractional derivative are studied. The equation holds in a time interval $(0,T)$ and it is assumed that a state $u$ (solution of diffusion equation) and a source $f$ are…
We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^{\gamma}}{\partial x^{\gamma}}[P(x,t) ]^{\nu}$. Exact time-dependent solutions are found for $ \nu =…
Anomalous relaxation and diffusion processes have been widely characterized by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to the singular memory kernel that…
Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either…
It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling…
Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\gamma>0$ and $\eta>0$. Suppose either (i) $\alpha\ne 0$ and $\beta=0$ or (ii) $\alpha\in\mathbb{R}$ and $\beta\ne 0$ holds. We will study the elliptic equation $\Delta (f^m/m)+\alpha f+\beta x\cdot\nabla…
We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…
This paper deals with an inverse source problem for the $1$D time-fractional diffusion equation by using boundary measurement. The conditional stability in identification of the unknown source term is proved on the basis of the Fourier…
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general…
This work considers the subdiffusion problem with non-positive memory, which not only arises from physical laws with memory, but could be transformed from sophisticated models such as subdiffusion or subdiffusive Fokker-Planck equation with…
Recently a new type of Kramers-Fokker-Planck Equation has been proposed [R. Friedrich et al. Phys. Rev. Lett. {\bf 96}, 230601 (2006)] describing anomalous diffusion in external potentials. In the present paper the explicit cases of a…
We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary…
Equation of long-range particle drift and diffusion on three-dimensional physical lattice is suggested. This equation can be considered as a lattice analogof space-fractional Fokker-Planck equation for continuum. The lattice approach gives…
Second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution…
A fractional diffusion equation based on Riemann-Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of $H$-functions. It differs from the known…
Anomalous diffusion phenomena are ubiquitous in complex media, such as biological tissues. A wide class of sub-diffusive phenomena phenomena is described by the time-fractional diffusion equation. The paper investigates the case of…
In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and…
We present in this paper a rigorous method to derive the nonlinear Fokker-Planck (FP) equation of anomalous diffusion directly from a generalization of the principle of least action of Maupertuis proposed by Wang for smooth or quasi-smooth…
We determine the space-dependent source term for a two-parameter fractional diffusion problem subject to nonlocal non-self-adjoint boundary conditions and two local time-distinct datum. A bi-orthogonal pair of bases is used to construct a…