Related papers: Method of Hidden Parameters and Pell's Equation
For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…
We obtain a new coding and decoding method using the generalized Pell $(p,i)$ -numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory. We give two new blocking…
The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: 1) if…
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…
Model selection methods are used in different scientific contexts to represent a characteristic data set in terms of a reduced number of parameters. Apparently, these methods have not found their way into the literature on multibody systems…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Optimization methods have been broadly applied to two classes of objects viz. (i) modeling and description of data and (ii) the determination of the stationary points of functions. Here, a theoretical basis is developed that optimizes an…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
In this chapter, we present some recent progresses on the numerics for stochastic distributed parameter control systems, based on the \emph{finite transposition method} introduced in our previous works. We first explain how to reduce the…
Explainability is needed to establish confidence in machine learning results. Some explainable methods take a post hoc approach to explain the weights of machine learning models, others highlight areas of the input contributing to…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
We propose an improved method for estimating partial differential equations and delay partial differential equations from data, using Bayesian optimization and the Bayesian information criterion to automatically find suitable…
The article addresses the convergence of implicit and semi-implicit, fully discrete approximations of a class of nonlinear parabolic evolution problems. Such schemes are popular in the numerical solution of evolutions defined with the…
A new initialization method for hidden parameters in a neural network is proposed. Derived from the integral representation of the neural network, a nonparametric probability distribution of hidden parameters is introduced. In this…
The numerical analysis for the small amplitude motion of an elastic beam with internal damping is investigated in domain with moving ends. An efficient numerical method is constructed to solve this moving boundary problem. The stability and…
In this paper, we investigate the parameter estimation problem for reflected OU processes. Both the estimates based on continuously observed processes and discretely observed processes are considered. The explicit formulas for the…