Yasutaka Sibuya

HOLOMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL SYSTEMS AT SINGULAR POINTS

Abstract : The paper gives a method for determining the existence of holomorphic solutions of linear differential systems. This method is similar to the method of determining (bifurcation) equations that have been used extensively in studying periodic differential equations. (Author)

A global analysis of matrices of functions of several variables

Let x = (x1, xs , 0-e) x,) be an r-dimensional vector, where q are real or complex numbers, and let 9 be the domain defined by I xj I < l (j = 1,2, *.., r). (l-1) Consider a family 3 of functions of x which are defined on 9, and consider an n by TZ matrix A(x) whose elements belong to this family 9. Let p(h; x) be the characteristic polynomial of A(x): p(h; x) = det (A(x) Al,), (1.2) where 1, is the TZ by TZ identity matrix, h is a complex variable, and det means the determinant. We assume that the polynomial p(X; X) is factored as PC& 4 = P&t x) PA& 4 for every x in 9, (1.3) where

The reciprocals of solutions of linear ordinary differential equations

On caracterise la classe des series de puissance qui avec leurs reciproques satisfont des equations differentielles lineaires

Formal power series solutions in a parameter

Abstract This paper is a continuation of the previous paper (J. Differential Equations 165 (2000) 255). The main subject is the Gevrey property of formal solutions of an analytic ordinary differential equation in powers of a parameter. In one case, a given formal solution itself is of the Gevrey type, while, in another case, the existence of a formal solution implies the existence of formal solutions of the Gevrey types. These situations are explained systematically in this paper.

Construction of a fundamental matrix solution at a singular point of the first kind by means of the SN decomposition of matrices

Abstract It is known that any matrix can be decomposed into a diagonalizable part and a nilpotent part. We call this the SN decomposition. We can derive the SN decomposition quite easily with a computer. Generalizing the SN decomposition to particular matrices of infinite order, we explain basic steps of construction of a linear transformation which reduces a given system of linear meromorphic ordinary differential equations to a normal form at a singular point of the first kind. Some examples are given utilizing Mathematica. We also show that the same idea produces a block-diagonalization of a given system at a singular point of the second kind.

The n-th Roots of Solutions of Linear Ordinary Differential Equations

In this paper we shall prove the following theorem: Let K be a differential field of characteristic zero. Let p and I be elements of a differential field extension of K such that (i) p # 0 and k # 0; (ii) p and k satisfy nontrivial linear differential equations with coefficients in K, say, P(y>) = 0 and Q(O) = 0; (iii) p = on for some positive integer n such that n > ord P. Then the logarithmic derivatives of p and 4 are algebraic over K. (Note that

CONVERGENCE OF POWER SERIES SOLUTIONS OF p-ADIC NONLINEAR DIFFERENTIAL EQUATION

Publisher Summary This chapter discusses convergence of power series solutions of p-adic nonlinear differential equation. It considers Qp, the field of rational p-adic numbers, and Ω, a complete algebraically closed extension of Qp. The Ord denotes the p-adic (additive) valuation in Ω that is normalized by the condition ord p = 1. In this chapter “convergence” means “convergence in the p-adic sense.” The chapter presents the theorem that states that if every root α of the indicial polynomial μq0(m) satisfies the condition ord(m + α) = 0(log m) as m → +∞ in Z, then, for any convergent power series f(z), every formal power series solution of the differential equation L(y) = f(z) is also convergent. An element α of Ω is called a p-adically non-Liouville number if the condition discussed is satisfied. The main purpose of this chapter is to generalize this theorem to a nonlinear differential equation. It also proves two lemmas on which the method of successive approximations is based.

SUBDOMINANT SOLUTIONS ADMITTING A PRESCRIBED STOKES PHENOMENON

Publisher Summary This chapter discusses subdominant solutions admitting a prescribed stokes phenomenon. It presents a general problem of constructing a system of linear differential equations that admits prescribed asymptotic behaviors of solutions and a prescribed Stokes phenomenon around an irregular singular point. Out-problem is different from that of Birkhoff, as, in given case, the prescribed asymptotic behavior of solutions depend on the parameters a1, …, am that must be determined by the special form of the differential equation and the prescribed Stotes phenomenon.

A note on partial differential-difference equation

In this paper, we studied a certain partial differential-difference equation which arose from crystal precipitation modeling. The uniqu existence and asymptotic behaviors of the solution were shown

Fundamental Theorems of Ordinary Differential Equations

In this chapter, we explain the fundamental problems of the existence and uniqueness of the initial-value problem