相关论文: Regularising the Cauchy problem for Laplace's equa…

This is the first publication in which an ill-posed Cauchy problem for a quasi- linear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is…

We construct solutions to p-Laplace type equations in unbounded Lipschitz domains in the plane with prescribed boundary data in appropriate fractional Sobolev spaces. Our approach builds on a Cauchy integral representation formula for…

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-\tau)u(\tau)\,d\tau -k(t)u(0)$ where $k$ is a…

This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight…

We introduce a class of fractional Dirac type operators with time variable coefficients by means of a Witt basis, the Djrbashian-Caputo fractional derivative and the fractional Laplacian, both operators defined with respect to some given…

The inverse problem of amplitude reconstruction on an inclined line based on the values of amplitude or its module as recorded on semi-infinite line orthogonal to the beam propagation direction is considered within the framework of 2D…

The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by…

In this article, we study the fractional Br\'{e}zis-Nirenberg type problem on whole domain $\mathbb{R}^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation*}…

The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $\Omega$, $u=f$ on $\partial\Omega$, from the…

In this paper we propose the design of an iterative observer using space as a time-like variable and prove its convergence. The iterative observer algorithm solves boundary estimation problem for a steady-state elliptic equation system…

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$-rectifiable boundary. In this paper we study the connection between the solvability of $(D_{p'})$,…

We investigate the Cauchy problem on the cylinder, namely the semi-periodic problem where there is periodicity in the $x$-direction and decay in the $y$-direction, for the Kadomtsev-Petviashvili II equation by the inverse spectral transform…

Duhamel's principle reduces the Cauchy problem for an inhomogeneous partial differential equation to the corresponding homogeneous problem. In the fractional-order setting, the classical principle does not apply directly because fractional…

Consider a path of the reflected Brownian motion in the half-plane $\{y \ge 0\}$, and erase its part contained in the interior $\{y > 0\}$. What is left is, in an appropriate sense, a path of a jump-type stochastic process on the line $\{y…

For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…

We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$ is a symmetric and coercive elliptic operator in $H^1(\Omega)$ with $\Omega$ bounded domain in ${{\mathbb R}}^d$. The boundary conditions involve fractional power…

Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of…

In this article, we initiate the study of the Cauchy problem for the two-dimensional relativistic Euler equations in a low-regularity setting. By introducing good variables--a rescaled velocity, logarithmic enthalpy, and an appropriately…

This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied…

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a…