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Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of…
The understanding and modeling of complex physical phenomena through dynamical systems has historically driven scientific progress, as it provides the tools for predicting the behavior of different systems under diverse conditions through…
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
It is shown, how to generate infinite sequences of differential equations of the second order based on some standard equations, using Euler-Imshenetsky-Darboux (EID) transformation. For all this, factorizations of differential operators and…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
We are interested in numerically approximating the solution ${\bf U}(t)$ of the large dimensional semilinear matrix differential equation $\dot{\bf U}(t) = { \bf A}{\bf U}(t) + {\bf U}(t){ \bf B} + {\cal F}({\bf U},t)$, with appropriate…
An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can…
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all…
A second-order Newton-based extremum seeking (SONES) algorithm is presented to estimate directional inflection points for multivariable static maps. The design extends the first-order Newton-based extremum seeking algorithm that drives the…
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…
A new method of solving third-order ordinary complex differential equations (OCDEs) by generalizing Prelle-Singer. The idea which is a procedure for finding the solution for second-order differential equations in the real domain. We have…
In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit…
Coupled second order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations we focus our attention on the…
A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the…
In this paper, we present a method of deriving extended Prelle-Singer method's quantifiers from Darboux Polynomials for third-order nonlinear ordinary differential equations. By knowing the Darboux polynomials and its cofactors, we extract…
We investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the…