Related papers: Fractional Cauchy problems on bounded domains
A family of dispersive equations is considered which links a higher dimensional Benjamin-Ono equation and the Zakharov-Kuznetsov equation. For these fractional Zakharov-Kuznetsov equations new well-posedness results are proved using…
In this paper, we study the Cauchy-Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \mbox{$\partial_t u - \operatorname{div} \left( D_\xi f(t, Du)\right) = 0$ } & \mbox{in $\Omega_T$}, \\[5pt] \mbox{$u = u_o$} & \mbox{on…
We study a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that…
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…
In this paper, we are interested in numerical solution of some linear boundary value problems with Dirichlet boundary part, by the means of simulation of random walks. We use a probabilistic interpretation of solution $u$, assuming that the…
We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized H\"older spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly…
In this paper, by using a Taylor development type formula, we show how it is possible to associate differential operators with stochastic differential equations driven by a fractional Brownian motion. As an application, we deduce that…
This paper is devoted to the study of the singularly perturbed second order partial integro-differential equations. The estimation of the solutions of Cauchy problem is obtained.
We study second-order stochastic parabolic equations in a cylindrical domain with homogeneous Dirichlet boundary conditions. Under a natural compatibility condition on the gradient-type noise, we establish global Schauder estimates in…
This article is devoted to developing an abstract theory of time-fractional gradient flow equations for time-dependent convex functionals in real Hilbert spaces. The main results concern the existence of strong solutions to time-fractional…
In this work we study partial differential equations defined in a domain that moves in time according to the flow of a given ordinary differential equation, starting out of a given initial domain. We first derive a formulation for a…
We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized…
This paper is concerned with the mathematical analysis of the inverse random source problem for the time fractional diffusion equation, where the source is assumed to be driven by a fractional Brownian motion. Given the random source, the…
In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time…
This paper is devoted to the fractional generalization of the Fokker-Planck equation associated with a stochastic differential equation in a bounded domain. The driving process of the stochastic differential equation is a L\'evy process…
Distributed order fractional model of viscoelastic body is used in order to describe wave propagation in infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem, corresponding to fractional wave…
This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion $B_H(t)$. We obtain solutions of these equations which are probability laws extending that of $B_H(t)$. Our analysis is…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the…
In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given…
We consider the Cauchy problem associated with a general parabolic partial differential equation in $d$ dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as…