Related papers: On polynomial solutions of differential equations
For the orthogonal Lie algebra O(2n+1), in addition to the conventional set of orthogonal polynomials, another set is produced with the help of the Lie superalgebra OSP(1|2n). Difficulties related with expression of Dyson's constant for the…
We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…
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…
A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
We introduce efficient differentially private (DP) algorithms for several linear algebraic tasks, including solving linear equalities over arbitrary fields, linear inequalities over the reals, and computing affine spans and convex hulls. As…
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…
A polynomial algorithm is obtained for the NP-complete linear ordering problem.
In recent years, sparse spectral methods for solving partial differential equations have been derived using hierarchies of classical orthogonal polynomials on intervals, disks, disk-slices and triangles. In this work we extend the…
A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes O(m^2n)…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary…
In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville…
We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth order linear differential equations, and one of the families is…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…