Related papers: On the existence of nonoscillatory phase functions…
We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately…
Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the…
It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the…
We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…
We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the…
In this paper we have considered higher order two dimensional coupled system of non-linear ordinary differential equations. We have given necessary and sufficient conditions on the non-linear functions such that the solutions pair oscilla
It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in…
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for…
We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic…
We obtain existence and uniqueness for odd second order oscillators in the space of odd functions without topological assumptions on the nonlinear part.
This paper systematically treats the asymptotic behavior of many (linear/nonlinear) classes of higher-order fractional differential equations with multiple terms. To do this, we utilize the characteristics of Caputo fractional…
In this letter we apply a method recently devised in \cite{aapla03} to find precise approximate solutions to a certain class of nonlinear differential equations. The analysis carried out in \cite{aapla03} is refined and results of much…
We report on the possibilities of using the method of normal fundamental systems for solving some problems of oscillation theory. Large elastic dynamical systems with continuous and discrete parameters are considered, which have many…
We introduce an efficient numerical method for second order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory…
We introduce a version of the asymptotic expansions for Bessel functions $J_\nu(z)$, $Y_\nu(z)$ that is valid whenever $|z| > \nu$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…
We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel…
New necessary and sufficient conditions are given for the quantization of a class of periodic second order non-homogeneous ordinary differential equations in the complex plane in this paper. The problem is studied from the viewpoint of…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…