Related papers: A Spectral Method for Solving the Cauchy Problem
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
Numerical relativity has traditionally been pursued via finite differencing. Here we explore pseudospectral collocation (PSC) as an alternative to finite differencing, focusing particularly on the solution of the Hamiltonian constraint (an…
This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized…
Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of L_{p} spaces of functions whose regularity is defined by a Levy measure with O-regulary varying radial profile. Existence and uniqueness of a solution…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems. It is assumed that constraint function values and derivatives are available, but only stochastic…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The mixed problem for a degenerate high order equation with a fractional derivative in a rectangular domain is considered in the article. The existence of a solution and its uniqueness are shown by the spectral method.
This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical…
The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…
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…
We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals $\mathrm{P}\!\int_{a}^{b} f(x)(x-\tau)^{-1} dx$ $(a < \tau < b)$ using standard adaptive quadratures. In order to properly…
A general method for solving nonlinear ill-posed problems is developed. The method consists of solving a Cauchy problem with a regularized operator and proving that the solution of this problem tends, as time grows, to a solution of the…
We present a novel solution method for It\^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main…
Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described…
The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gauss-type quadrature formula is used to approximate integrals during the…