Related papers: Singular Initial Value Problems for Scalar Quasi-L…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
The method of constructing approximate solutions of the first boundary value problem for linear differential equations based on incomplete (even and odd) trigonometric splines is considered. The theoretical positions are illustrated by…
In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that…
We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and…
In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial…
Accurate initial conditions have the task of precisely capturing and fixing the free integration constants of the flow considered. This is trivial for regular ordinary differential equations, but a complex problem for differential-algebraic…
We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the…
First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial…
We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] \times \Sigma$, where $\Sigma$ is a compact manifold with smooth boundaries $\partial\Sigma$. By using an appropriate…
In this work, we study the initial value problem associated with an abstract integrodifferential equation in interpolation scales. We prove local-in-time existence, uniqueness, continuation, and a blow-up alternative for regular mild…
We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak…
Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an…
In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition $$\alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma,$$ for some positive constants $\alpha$, $\beta$, and $\sigma$,…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…
This paper investigates the initial boundary value problem for a fractional pseudo-parabolic equation with singular potential. The global existence and blow-up of solutions to the initial boundary value problem are obtained at low initial…
In this article we will study the initial value problem for some Schr\"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in…
We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions…
We consider the initial boundary value problem for the focusing nonlinear Schr\"odinger equation in the quarter plane $x>0,t>0$ in the case of decaying initial data (for $t=0$, as $x\to +\infty$) and the Robin boundary condition at $x=0$.…
In this letter we are concerned with the possibility to approach the existence of solutions to a class of discontinuous dynamical systems of fractional order. In this purpose, the underlying initial value problem is transformed into a…
In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in…