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We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard…
Super-resolution is generally referred to as the task of recovering fine details from coarse information. Motivated by applications such as single-molecule imaging, radar imaging, etc., we consider parameter estimation of complex…
This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $\epsilon$-expansion series with numerical coefficients. The algorithm is based on…
Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and…
This paper exhibits a very simple formula for a particular solution of a linear ordinary differential equation with constant real coefficients, P(d/dt)x = f, f a function given by a linear combination of polynomials, trigonometrical and…
We look at the number of solutions of an equation of the form f_1*f_2*...*f_k=a in a finite field, where each f_i is a multilinear polynomial. We use two methods to construct a solution of this problem for the cases a=0, a<>0, and we…
In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
Solving Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions. However, few work about global error estimation can be…
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…
We provide algorithms computing power series solutions of a large class of differential or $q$-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
The general decomposition theory of exponential operators is briefly reviewed. A general scheme to construct independent determining equations for the relevant decomposition parameters is proposed using Lyndon words. Explicit formulas of…
Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the last years. However, the polynomial case has not been…