Related papers: On well-posedness of vector-valued fractional diff…

We show new results of wellposedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. For the purpose, we propose two approaches on the basis of the contraction-mapping argument. One of them relies upon the…

Given a linear closed but not necessarily densely defined operator $A$ on a Banach space $E$ with nonempty resolvent set and a multivalued map $F\colon I\times E\map E$ with weakly sequentially closed graph, we consider the…

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $ D^{\alpha}_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha\le1, ( *) $ where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in the…

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously…

Given sufficiently regular data \textit{without} decay assumptions at infinity, we prove local well-posedness for non-linear dispersive equations of the form \[ \partial_t u + \mathsf A(\nabla) u + \mathcal Q(|u|^2) \cdot \nabla u= \mathcal…

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the source $f$ also lies in (different) weighted tent spaces, provided the complex…

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…

In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and…

In this paper, we study the $\ell^p$-maximal regularity for the fractional difference equation with finite delay: \begin{equation*} \ \ \ \ \ \ \ \ \left\{\begin{array}{cc} \Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n), \ n\in \mathbb…

Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential…

In this paper we consider fractional quasi-Bessel equations $$\sum_{i=1}^{m}d_i x^{\alpha_i+p_i}D^{\alpha_i} u(x) + (x^\beta - \nu^2)u(x)=0$$ and construct their existence and uniqueness theory in the class of fractional series. Our…

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure \[ \partial_t u - \div\big(M\nabla u+ E h(u)\big)=0, \] with $M$ a bounded elliptic matrix, $E$ a vector field in a suitable Lebesgue…

We study existence and uniqueness of bounded solutions to a fractional nonlinear porous medium equation with a variable density, in one space dimension.

We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully…

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…

The survey is devoted to numerical solution of the fractional equation $A^\alpha u=f$, $0 < \alpha <1$, where $A$ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain…

We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie…

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-\Delta)^{\sigma/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt]…

Fractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order…

Operator splitting methods allow to split the operator describing a complex dynamical system into a sequence of simpler subsystems and treat each part independently. In the modeling of dynamical problems, systems of (possibly coupled)…