Construction Of Formal Solutions Research Articles

Formal and Locally Analytic Solutions of an Inhomogeneous Linear Functional Equation

For a given function \(\psi \) where the order of \(\psi \) is greater than one, \(a \in \mathbb {C}{\setminus \!} \left\{ 0 \right\} \) and \(k \ge 2\), we study locally analytic solutions h of the functional equation \(az^k+ (1+z^k)h(z)=h(\psi (z))\) with \(h(0)= w_0 \in \mathbb {C}\). After the construction of formal solutions, we discuss their convergence. Also, a representation of the solutions by infinite products is obtained.

Principally tame regular sequences associated with the fourth Painlevé hierarchy with a large parameter

The notion of principally tame regular sequences is introduced for systems of polynomials with a weight vector. As an application, construction of formal solutions is discussed for the systems of nonlinear differential equations which belong to the fourth Painleve hierarchy with a large parameter.

Analytical Proper Elements for the Hilda Asteroids I: Construction of a Formal Solution

In this paper, we present the mathematical basis for the calculation of proper elements for asteroids in 3:2 mean-motion resonance with Jupiter from their osculating Keplerian elements. The method is based on a new resonant Lie-series perturbation theory (Ferraz-Mello, 1997, 2002), which allows the construction of formal solutions in cases where resonant and long-period angles appear simultaneously. For the disturbing function, we used the Beauge’s expansion (Beauge, 1996), adapted to include short period terms. In this paper, the theory is restricted to the planar case and only the perturbations due to Jupiter are considered.

Singular-perturbative Reduction to BIrkhoff Normal Form and Instanton-Type Formal Solutions of Hamiltonian Systems

As is illustrated by the computation of monodromy groups of second-order Fuchsian equations (cf. [AKT1]), the exact WKB analysis provides us with a powerful tool for studying global behavior of solutions of linear ordinary differential equations. To generalize such an analysis to nonlinear equations, T. Kawai (RIMS, Kyoto Univ.), T. Aoki (Kinki Univ.) and the author have developed the WKB theory for Painleve equations with a large parameter in our series of articles ([KT1], [AKT2], [KT2]). (See [Tl], [T2] also.) In our treatment 2-parameter formal solutions called instanton-type solutions, which were constructed through the multiple-scale analysis in [AKT2], are playing a central role. Although we have succeeded in analyzing their local structure near simple turning points in [KT2], some of their important properties such as the behavior near fixed singular points have not been clarified yet. In this paper, to investigate their behavior near fixed regular-type singular points, we propose a new construction of 2-parameter formal solutions of Painleve equations with a large parameter. The new construction of formal solutions we propose here is based on Takano's work [Tkal] (see [Tka2] also), where he constructed a 2-parameter family of analytic solutions at each regular-type singular point of (ordinary) Painleve equations. He made use of the well-known fact that Painleve equations can be written in the form of Hamiltonian systems (which we call Painleve Hamiltonian systems here) and established some reduction theorem for

Construction of formal solutions of a linear system of integrodifferential equations with small parameter and variable delay

Construction of formal solutions of a linear system of integrodifferential equations with small parameter and variable delay