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The article is dedicated towards the study of fractional order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions…
The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \& R\"ockner, M., A mild Ito formula for SPDEs, arXiv:1009.3526 (2012), To appear in the Trans.\ Amer.\ Math.\ Soc.] has turned out to be a useful instrument to study…
We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study…
This paper deals with the initial value problem for the multi-term fractional differential equation. The fractional derivative is defined in the Caputo sense. Firstly the initial value problem is transformed into a equivalent Volterra-type…
This paper establishes integral representations of mild solutions of impulsive Hilfer fractional differential equations with impulsive conditions and fluctuating lower bounds at impulsive points. Further, the paper provides sufficient…
This article is devoted to the study of the existence and uniqueness of mild solution to time- and space-fractional stochastic Burgers equation perturbed by multiplicative white noise. The required results are obtained by stochastic…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
This paper deals with the Cauchy-Dirichlet problem for the fractional Cahn-Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper…
This paper investigates the Cauchy problem of the time-space fractional Keller-Segel-Navier- Stokes model, which can describe both memory effect and L\'evy process of the system. The local existence and global existence in Lebesgue space…
Our aim in this paper is to establish stable manifolds near hyperbolic equilibria of fractional differential equations in arbitrary finite dimensional spaces.
In the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost…
The main objective of this paper is to study the existence of solutions to some basic fractional difference equations. The tools employed are Krasnosel'skii fixed point theorem which guarantee at least two positive solutions.
We develop a regularity theory for integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge-Amp\`{e}re equation. We prove an ABP estimate and a Harnack inequality and derive H\"{o}lder and…
We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations in which the classical non-local Eringen constitutive…
In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in $\mathbb{R}^{n}$ for $n=2,\,3$. In comparison with the work of the 3D fractional Navier-Stokes equations obtained…
Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schr\"odinger equations are key to the intersection of nonlinear dynamics and fractional calculus. In this manuscript, the first discrete/differential…
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of…
In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers…
The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…
Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with Bounded Deformation. Mild additional continuity assumptions…