Related papers: Algebraic Approaches to Partial Differential Equat…
Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis, providing a framework to model and exploit structural and geometric properties inherent in visual data. Over the years,…
In this paper, we use various anstazes motivated from our earlier works on transonic gas flows, boundary layer problems and Navier-Stokes equations to find new explicit exact solutions with multiple parameter functions for the equation of…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide…
Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed…
In this work, we generalize the reaction-diffusion equation in statistical physics, Schr\"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics into the neural partial differential equations (NPDE), which can be…
We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing…
The paper presents a reformulation of some of the most basic entities and equations of linear elasticity - the stress and strain tensor, the Cauchy Navier equilibrium equations, material equations for linear isotropic bodies - in a modern…
Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…
Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic.…
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.…
A very simple and accurate numerical method which is applicable to systems of differentio-integral equations with quite general boundary conditions has been devised. Although the basic idea of this method stems from the Keller Box method,…
The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg--de Vries, coupled KdV, Harry…
Proposed in this paper is a numerical procedure to generate periodic traveling wave solutions of some nonlinear dispersive wave equations. The method is based on a suitable modification of a fixed point algorithm of Petviahvili type and…
In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some…
We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…