Related papers: Remarks on a fractional-time stochastic equation
For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise,…
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
Inspired by the works of \cite{baz2} and \cite{kian}, this study develops an abstract framework for analyzing differential equations with space-dependent fractional time derivatives and bounded operators. Within this framework, we establish…
This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of…
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 $D\subset\mathbb{R}^d$…
Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space-time fractional…
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 provide explicit classical solutions and stochastic analogues for distributed-order space-time fractional diffusion equations on bounded domains with zero exterior boundary conditions. We also show that our results still hold when the…
We consider fractional diffusion equations and study the stability of the inverse problem of determining the time-dependent parameter in a source term or a coefficient of zero-th order term from observations of the solution at one point in…
For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the…
In this paper we study the long time behavior of the solution to a certain class of space-time fractional stochastic equations with respect to the level $\lambda$ of a noise and show how the choice of the order $\beta \in (0, \,1)$ of the…
In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. The fractional derivative models time delays in a diffusion process. The order of the fractional derivative can be distributed…
The Schrodinger equation is considered with the first order time derivative changed to a Caputo fractional derivative, the time fractional Schrodinger equation. The resulting Hamiltonian is found to be non-Hermitian and non-local in time.…
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity.…
In this work, we introduce a novel variational framework for the study of the unsteady Stokes equations in a bounded open Lipschitz domain in R^n, involving a Caputo fractional derivative in time. The nonlocal nature of the fractional…
This paper presents a better approach to model an engineering problem in fractal-time space based on local fractional calculus. Some examples are given to elucidate to establish governing equations with local fractional derivative.
We consider the time-fractional Cattaneo equation involving the tempered Caputo space-fractional derivative. We find the characteristic function of the related process and we explain the main differences with previous stochastic treatments…
This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and…
In this paper we investigate the porous medium equation with a fractional temporal derivative. We justify that the resulting equation emerges when we consider the waiting-time (or trapping) phenomenon that can happen in the medium. Our…
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…