Related papers: Fractional Supershifts and their associated Cauchy…
We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…
We consider the Cauchy problem for stochastic fractional evolution equations with Caputo time fractional derivative of order $1<\alpha<2$ and space variable coefficients on an unbounded domain. The space derivatives that appear in the…
We describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order $\alpha \in (0,1)$ in the time variable $t$ and the first order derivatives in spatial variables…
In this paper we introduce a Fock space related to derivatives of Gelfond-Leontiev type, a class of derivatives which includes many classic examples like fractional derivatives or Dunkl operators. For this space we establish a modified…
We study a space-fractional Stefan problem with the Dirichlet boundary conditions. It is a model that describes superdiffusive phenomena. Our main result is the existence of the unique classical solution to this problem. In the proof we…
This is a first version of a paper concerning abstract evolution equation with fractional time derivatives. Maximal regularity results in spaces of continuous and Hoelder continuous functions are described.
In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by \citet{nigmatullin}, and \citet{zaslavsky} in $\mathbb R^d$ for modeling some physical…
We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $|x|$…
We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of ``stochastic composition" of the solutions to two simpler problems. These Cauchy sub-problems respectively…
We investigate evolution equations for anomalous diffusion employing fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. Fractional diffusion equations…
We consider the Cauchy problem for a degenerate fractional conservation laws driven by a noise. In particular, making use of an adapted kinetic formulation, a result of existence and uniqueness of solution is established. Moreover, a…
A class of linear evolutionary equations with material laws involving fractional time-derivatives is considered. The main result is well-posedness and causality for this problem class. The approach is illustrated with two examples: a…
We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the…
The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann-Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space.…
We develop a theory of the Cauchy problem for linear evolution systems of partial differential equations with the Caputo-Dzrbashyan fractional derivative in the time variable $t$. The class of systems considered in the paper is a fractional…
Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\subset\mathbb{R}^d$…
In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
We establish trace and extension theorems for evolutionary equations with the Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we identify weighted Sobolev and Besov spaces with mixed norms that accommodate…
Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases…