Related papers: How much can one learn from a single solution of a…
In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…
In this letter we obtain sharp estimates on the growth rate of solutions to a nonlinear ODE with a nonautonomous forcing term. The equation is superlinear in the state variable and hence solutions exhibit rapid growth and finite-time…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external…
This paper considers the growth rates of positive solutions of scalar nonlinear functional and Volterra differential equations. The equations are assumed to be autonomous (or asymptotically so), and the nonlinear dependence grows less…
Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…
This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows…
The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems…
Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…
We consider a linear equation $\partial_t u = \mathcal{L}u$, where $\mathcal{L}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field…
This chapter focuses on variable maturation delay or, more precisely, on the mathematical description of a size-structured population consuming an unstructured resource. When the resource concentration is a known function of time, we can…
In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's…
The paper introduces a PDE model for the growth of a tree stem or a vine. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles. An additional term accounts for the tendency of a…
We study the semilinear elliptic equation $\Delta u + g(x,u,Du) = 0$ in $\R^n$. The nonlinearities $g$ can have arbitrary growth in $u$ and $Du$, including in particular the exponential behavior. No restriction is imposed on the behavior of…
The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
We consider the problem of finding $\lambda\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{\lambda + F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic,…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=\lambda\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter.…
The realistic modeling intended to quantify precisely some biological mechanisms is a task requiering a lot of a priori knowledge and generally leading to heavy mathematical models. On the other hand, the structure of the classical Machine…
We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…