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We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
We derive sharp decay estimates and prove holomorphic extensions for the solutions of a class of semilinear nonlocal elliptic equations with linear part given by a sum of Fourier multipliers with finitely smooth symbols at the origin.…
In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to variational approach in the general case we have the solution as a…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp…
This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear…
This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.
The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate…
In this paper, approximate solutions for a class of fractional Lane - Emden type equations based on the series expansion method are presented. Various examples are introduced and discussed. The recurrence relation for the components of the…
In order to find analytically the travelling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: "projective Riccati method", "tanh-method", "exponential method",…
The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…
Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are…
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated…
In this paper, we study the global existence and regularity of H\"older continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and…
We discuss a version the methodology for obtaining exact solutions of nonlinear partial differential equations based on the possibility for use of: (i) more than one simplest equation; (ii) relationship that contains as particular cases the…
We derive parametric travelling-wave solutions of non-linear QCD equations. They describe the evolution towards saturation in the geometric scaling region. The method, based on an expansion in the inverse of the wave velocity, leads to a…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…