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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 consider the Cauchy problem for a second-order evolution equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the…

Let E be a type 2 UMD Banach space, H a Hilbert space and let p be in [1,\infty). Consider the following stochastic delay equation in E: dX(t) = AX(t) + CX_t + b(X(t),X_t)dW_H(t), t>0; X(0) = x_0; X_0 = f_0. Here A : D(A) -> E is the…

The basic aim is to extend some results and concepts of non-autonomous second order differential systems with convex potentials to the new context of multi-time Poisson-gradient PDE systems with convex potential. In this sense, we prove…

In this paper we investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}%…

In this paper, a general theorem on the equivalence of pth moment stability between stochastic differential delay equations (SDDEs) and their numerical methods is proved under the assumptions that the numerical methods are strongly…

By observing that the fractional Caputo derivative can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo…

We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in…

We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of…

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract…

We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…

We obtain a vector-valued subordination principle for $(g_{\alpha}, g_{\alpha})$-regularized resolvent families which unified and improves various previous results in the literature. As a consequence we establish new relations between…

We consider a rather general class of evolutionary PDEs involving dissipation (of possibly fractional order), which competes with quadratic nonlinearities on the regularity of the overall equation. This includes as prototype models,…

Extending the results of Nardi (2015), this note establishes an existence and uniqueness result for second-order uniformly elliptic PDEs in divergence form with Neumann boundary conditions. A Schauder estimate is also derived.

For the linear partial differential equation $P(\partial_x,\partial_t)u=f(x,t)$, where $x\in\mathbb{R}^n,\;t\in\mathbb{R}^1$, with $P(\partial_x,\partial_t)$ is $\prod^m_{i=1}(\frac{\partial}{\partial{t}}-a_iP(\partial_x))$ or…

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under the weakly local Lipschitz and some suitable conditions, a generic truncated Euler-Maruyama (TEM) scheme for SDDEs is…

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper sub-diffusive cases are…

A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…

The abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order $\beta\in(0,1)$ and operator $-A^\alpha$, $\alpha\in(0,1)$, is considered, where $-A$ generates a strongly continuous one-parameter…

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…