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We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older derivatives (for some $0<\alpha\leq 1$). The smooth approximation is given by means of an…
Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example,…
Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets…
We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to…
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 paper considers the approximation of spatial convolution with a given radial integral kernel. Previous studies have demonstrated that approximating spatial convolution using a system of partial differential equations (PDEs) can…
Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the form $f(I_1(u),\ldots,I_k(u))\geq 0$ where $f$ is a polynomial, $u$ is any function satisfying prescribed…
Motivated by problems where the response is needed at select localized regions in a large computational domain, we devise a novel finite element discretization that results in exponential convergence at pre-selected points. The two key…
We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the…
Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…
Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations…
Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals,…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
For the first time, a nonlinear Schr\"odinger equation of the general form is considered, depending on time and two spatial variables, the potential and dispersion of which are specified by two arbitrary functions. This equation naturally…
We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including…
We present and analyse a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we…
In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs…
In this article we develop convergence theory for a general class of adaptive approximation algorithms for abstract nonlinear operator equations on Banach spaces, and use the theory to obtain convergence results for practical adaptive…