Related papers: Error analysis for time-fractional semilinear para…
A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying…
In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $\alpha \in (0, 1)$. As initial data, we consider a pair consisting of an initial point, which does not…
We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data…
In this paper, we discretize the Caputo time derivative of order \alpha \in (0,1) using the Alikhanov scheme on a quasi-graded temporal mesh, and employ the Newton linearization method to approximate the nonlinear term. This yields a…
We study a semilinear fractional-in-time Rayleigh-Stokes problem for a generalized second-grade fluid with a Lipschitz continuous nonlinear source term and initial data $u_0\in\dot{H}^\nu(\Omega)$, $\nu\in[0,2]$. We discuss stability of…
We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \begin{equation}\label{0.1} 0\leq(\partial_t-\Delta)^\alpha u\leq u^\lambda \quad\text{ in }\mathbb{R}^n…
For the numerical solution of Dirichlet-type boundary value problems associated to nonlinear fractional differential equations of order $\alpha \in (1,2)$ that use Caputo derivatives, we suggest to employ shooting methods. In particular, we…
A class of linear parabolic equations is considered. We derive a framework for the a posteriori error analysis of time discretisations by Richardson extrapolation of arbitrary order combined with finite element discretisations in space. We…
This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known…
Computable estimates for the error of finite element discretisations of parabolic problems in the $L^\infty(0,T; L^2)$ norm are developed, which exhibit constant effectivities (the ratio of the estimated error to the true error) with…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality…
In the paper we deal with linear fractional control problems with constant delays in the state. Single-order systems with fractional derivative in Caputo sense of orders between 0 and 1 are considered. The aim is to introduce a new…
Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods (MFEM) in space for simulating…
In this paper we investigate existence of solutions for the system: \begin{equation*} \left\{ \begin{array}{l} D^{\alpha}_tu=\textrm{div}(u \nabla p),\\ D^{\alpha}_tp=-(-\Delta)^{s}p+u^{2}, \end{array} \right. \end{equation*} in…
In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori…
We study some inverse problems for time-fractional Schr\"odinger equations involving the Caputo derivative of fractional order $\alpha \in (0,1)$. We prove refined uniqueness results from sets of positive Lebesgue measure for various…
The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type…
In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e., an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the…
We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order~$\alpha \in (0,1)$. The basic idea of our scheme is based on local integration followed by linear…
We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates…