Related papers: A solution of polynomial equations
A class of bivariate infinite series solutions of the elliptic and hyperbolic Kepler equations is described, adding to the handful of 1-D series that have been found throughout the centuries. This result is based on an iterative procedure…
Numerical methods for the Euler equations with a singular source are discussed in this paper. The stationary discontinuity induced by the singular source and its coupling with the convection of fluid presents challenges to numerical…
In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More…
A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…
Iterative equation is an equality with an unknown function and its iterates. There were not found a result on iterative equations with multiplication of iterates of the unknown function on $\mathbb{R}$. In this paper we use an exponential…
A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from…
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…
In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
To the best of our knowledge this paper is the first attempt to introduce and study polynomial interpolation of the polynomial data given on arbitrary varieties. In the first part of the paper we present results on the solvability of such…
This chapter deals with the exact enumeration of certain classes of self-avoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of…
We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables.…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
We study the modular resolution method using new tools called polynumbers and polyseries, introduced by Prof. Wildberger N.J. We try to prove an equivalence theorem of the existence and the uniqueness of the solutions of the modular…
In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
In this paper a new approach is derived in the context of shape theory. The implemented methodology is motivated in an open problem proposed in \citet{GM93} about the construction of certain shape density involving Euler hypergeometric…
We consider resultant-based methods for elimination of indeterminates of Ore polynomial systems in Ore algebra. We start with defining the concept of resultant for bivariate Ore polynomials then compute it by the Dieudonne determinant of…