相关论文: Linear Differential Equations and Orthogonal Polyn…

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…

In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials.…

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…

A method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points. This yields a robust and efficient method which can avoid singularities and breakdowns in the…

Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that…

Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for…

Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational,…

In the present work, an attempted was made to develop a numerical algorithm by the use of new orthogonal hybrid functions formed from hybrid of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal…

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…

A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…

A finite element method for solving nonlinear differential equations on a grid, with potential applicability to computational fluid dynamics (CFD), is developed and tested. The current method facilitates the computation of solutions of a…

The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is…

In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of…

In this paper we introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of…

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained…

We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…

We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and…

A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established…