Related papers: Cauchy non-integral formulas
We solve the Cauchy problem defined by the fractional partial differential equation $[\partial_{tt}-\kappa\mathbb{D}]u=0$, with $\mathbb{D}$ the pseudo-differential Riesz operator of first order, and the initial conditions…
This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for data $\frac{1}{4}$ derivatives less…
In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data…
Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method…
We review the integration of the KP hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions…
We consider the Calder\'on problem in the case of partial Dirichlet-to-Neumann map for the system of elliptic equations in a bounded two dimensional domain. The main result of the manuscript is as follows: If two systems of elliptic…
The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo first constructed a global smooth irrotational solution…
Two methods of level set type are proposed for solving the Cauchy problem for an elliptic equation. Convergence and stability results for both methods are proven, characterizing the iterative methods as regularization methods for this…
In this paper, we establish a sharp $C^{2+\alpha}$-theory for stochastic partial differential equations of parabolic type in the whole space.
We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the…
Two novel extended semi-discrete KP-type systems, namely partial differential-difference systems with one continuous and two discrete variables, are investigated. Introducing an arbitrary function into the Cauchy matrix function or the…
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati…
This paper is the continuation of the paper arXiv:1509.06950, which is Part I under the same title. In this paper, we prove a generalized Cauchy formula for the integrals of logarithmic forms on products of projective lines, and give an…
We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada…
We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local…
The study of the Dirac system and second-order elliptic equations with complex-valued coefficients on the plane leads to bicomplex Vekua equations. To the difference of complex pseudoanalytic (generalized analytic) functions the theory of…
In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\nu k$, for $k$ non-negative integer and $\nu>0$. The solution is obtained by connecting the…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…
The purpose of this paper is to obtain an upper bound for the fundamental solution for parabolic Cauchy problem u'=Au, where A is a second order elliptic partial differential operator with unbounded coefficients such that its potential and…
The well-posedness of the abstract \textsc{Cauchy} problem for the doubly nonlinear evolution inclusion equation of second order \begin{align*} \begin{cases} u''(t)+\partial \Psi(u'(t))+B(t,u(t))\ni f(t), &\quad t\in (0,T),\, T>0,\\…