Related papers: Multiplicity of Solutions for Linear Partial Diffe…
The concept of "multiplicity of solutions" was developed in arXiv:1509.02603v2 which is based on the theory of energy operators in the Schwartz space S^-(R) and some subspaces called energy spaces first defined in arXiv:1208.3385 and…
This work starts with the introduction of a family of differential energy operators. Energy operators ($Psi_R^+$, $Psi_R^-$) were defined together with a method to decompose the wave equation in a previous work. Here the energy operators…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
This is a survey paper based on previous results of the author. In the paper, we define and discuss the generalizations of linear partial differential equations to multidimensional variational problems. We consider two examples of such…
In this work we propose a hybrid solver to solve partial differential equation (PDE)s in the latent space. The solver uses an iterative inferencing strategy combined with solution initialization to improve generalization of PDE solutions.…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
We consider the total energy decay together with L^2-bound of the solution itself of the Cauchy problem for wave equations with a localized damping and a short-range potential. We treat it in the one dimensional Euclidean space R. We adopt…
This work is concerned with the quantification of the epistemic uncertainties induced the discretization of partial differential equations. Following the paradigm of probabilistic numerics, we quantify this uncertainty probabilistically.…
The non-commutative strategy developed by Bagarello (see Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394) for the analysis of systems of ordinary differential equations (ODEs) is extended to a class of partial…
A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential…
This paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel. Method allows one to construct the systems of multidimensional Partial Differential…
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of…
Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by…
In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling critical examples. By introducing Gelfand triples and…
This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial…
It has been found a simple procedure for the general solution of the time-independent Schr\"odinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not…
The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these…
Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data.…
We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…