A Relation between formal and Asymptotic Equivalence for linear Systems Containing A Parameter

In §§V-1 and V-2, we defined formal solutions of a system of analytic differential equations. Formal solutions are not necessarily convergent. For example, as we mentioned it in Remark V-1-4, the divergent formal power series \(\hat{f} = \sum\limits_{{m = 0}}^{\infty } {{{{\left( { - 1} \right)}}^{m}}} \left( {m!} \right){{x}^{{m + 1}}}\) is a formal solution of \({{x}^{2}}\frac{{dy}}{{dx}} + y - x = 0\). This equation has an actual solution \(f\left( x \right) = {{e}^{{{{1} \left/ {x} \right.}}}}\int_{0}^{x} {{{t}^{{ - 1}}}{{e}^{{{{{ - 1}} \left/ {t} \right.}}}}} dt{\text{ for }}x > 0\) Integrating by parts,we obtain \(f = \sum\limits_{{m = 0}}^{N} {{{{\left( { - 1} \right)}}^{m}}\left( {m!} \right){{x}^{{m + 1}}} + {{{\left( { - 1} \right)}}^{{N + 1}}}\left( {\left( {N + 1} \right)!} \right){{e}^{{{{1} \left/ {x} \right.}}}}\int_{0}^{x} {{{t}^{N}}{{e}^{{{{{ - 1}} \left/ {t} \right.}}}}} dt}\) Since \(0 < {{e}^{{{{1} \left/ {x} \right.}}}}\int_{0}^{x} {{{t}^{N}}{{e}^{{{{{ - 1}} \left/ {t} \right.}}}}dt = {{x}^{{N + 2}}} - \left( {N + 2} \right){{e}^{{{{1} \left/ {x} \right.}}}}\int_{0}^{x} {{{t}^{{N + 1}}}{{e}^{{{{{ - 1}} \left/ {t} \right.}}}}dt < {{x}^{{N + 2}}}} }\) we conclude that \(\left| {f\left( x \right) - \sum\limits_{{m = 0}}^{N} {{{{\left( { - 1} \right)}}^{m}}\left( {m!} \right){{x}^{{m + 1}}}} } \right| < \left( {\left( {N + 1} \right)!} \right){{x}^{{N + 2}}}\) an asymptotic representation of an actual solution by means of a formal solution. In this chapter, we explain the asymptotic expansions of functions in the sense of Poincaré and in the sense of the Gevrey asymptotics. In the Poincaré asymptotics, flat functions are characterized by the condition \(\mathop{{\lim }}\limits_{{x \to 0}} \frac{{f\left( x \right)}}{{{{x}^{m}}}} = 0\), where c, k, and M are some positive numbers. Generally speaking, the Poincaré asymptotics is too general for the study of ordinary differential equations. A motivation of the Gevrey asymptotics is also given by the Maillet Theorem (cf. Theorem V-1-5). In §XI-1, we summarize the basic properties of asymptotic expansions of functions in the sense of Poincaré. The Gevrey asymptotics is explained in §§XI-2-XI-5.KeywordsLine SegmentPower SeriesAsymptotic ExpansionFormal SolutionFormal Power SeriesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this chapter, we consider a system of differential equations $$x\frac{{d\vec{y}}}{{dx}} = \vec{f}\left( {x,\vec{y}} \right)$$ (E) assuming that the entries of the \({{\mathbb{C}}^{n}}\)-valued function \(\vec{f}\) are convergent power series in complex variables \(\left( {x,\vec{y}} \right) \in {{\mathbb{C}}^{{n + 1}}}\) with coefficients in ℂ, where x is a complex independent variable and \(\vec{y} \in {{\mathbb{C}}^{n}}\) is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed power series solutions. However, generally speaking, in order to construct a power series solution \(\vec{y}\left( x \right) = \sum\limits_{{m = 1}}^{\infty } {{{x}^{m}}{{{\vec{a}}}_{m}}}\) this expression is inserted into the given differential equation to find relationships among the coefficients \({{\vec{a}}_{m}}\), and the coefficients \({{\vec{a}}_{m}}\) are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate \(|{{\vec{a}}_{m}}|\) to test its convergence. As the function \({{x}^{{ - 1}}}\vec{f}\left( {x,\vec{y}} \right)\) is not analytic at x = 0, Theorem I-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119] and [Wasl, Theorem 5.3, pp. 22-25] for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator $$ \mathcal{L}\left[ {\vec y} \right]{\text{ = }}x\frac{{d\vec y}}{{dx}}{\text{ + }}\Omega {\text{(}}x{\text{)}} \vec y, $$ (LDO) Where \(\Omega \left( x \right)\) is an n x n matrix whose entries are formal power series in x with coefficients in ℂ n . In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator L by using a method due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-89]). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].

We study a class of nonlinear difference equations admitting a 1-Gevrey formal power series solution which, in general, is not 1- (or Borel-) summable. Using right inverses of an associated difference operator on Banach spaces of so-called quasi-functions, we prove that this formal solution can be lifted to an analytic solution in a suitable domain of the complex plane and show that this analytic solution is an accelero-sum of the formal power series.

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f(x,u,D_x u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(

On the summability of divergent power series satisfying singular PDEs

The Gevrey asymptotics in the initial value problem for singularly perturbed nonlinear differential equations

We consider a singularity perturbed nonlinear differential equation εu ′ =f(x)u++εP(x,u,ε) which we suppose real analytic for x near some interval [a,b] and small |u|, |ε|. We furthermore suppose that 0 is a turning point, namely that xf(x) is positive if x≠0. We prove that the existence of nicely behaved (as ϵ→0) local (at x=0) or global, real analytic or C ∞ solutions is equivalent to the existence of a formal series solution ∑u n (x)ε n with u n analytic at x=0. The main tool of a proof is a new "principle of analytic continuation" for such "overstable" solutions. We apply this result to the second order linear differential equation εy ′′ +φ(x,ε)y ′ +ψ(x,ε)y=0 with φ and ψ real analytic for x near some interval [a,b] and small |ε|. We assume that -xφ(x,0) is positive if x≠0 and that the function ψ 0 :x↦ψ(x,0) has a zero at x=0 of at least the same order as φ 0 ↦φ(x,0). For this equation, we prove that the existence of local or global, real analytic or C ∞ solutions tending to a nontrivial solution of the reduced equation φ(x,0)y ′ +ψ(x,0)y=0 is equivalent to the existence of a non trivial formal series solution y ^(x,ε)=∑y n (x)ε n with y n analytic at x=0. This improves and generalizes a result of C.H. Lin on this so-called " Ackerberg-O’Malley resonance" phenomenon. In the proof, the problem is reduced to the preceding problem for the corresponding Riccati equation In the final section, we construct examples of such second order equations exhibiting resonance such that the formal solution y ^ has a prescribed logarithmic derivative y ^ ′ (0,ε)/y ^(0,ε) at x=0 which is divergent of Gevrey order 1.

Formal power series and regular operations on fuzzy languages

Strongly Hopfian Rings of Formal Power Series and Power Series Armendariz Rings in Infinitely Many Variables

Joseph Becker and Leonard Lipshitz. Remarks on the elementary theories of formal and convergent power series. Fundament a mathematicae, vol. 105 (1980), pp. 229–239. - Françoise Delon. Indécidabilité de la théorie des anneaux de séries formelles à plusiers indéterminées. Fundament a mathematicae, vol. 112 (1981), pp. 215–229. - J. Becker, J. Denef, and L. Lipshitz. Further remarks on the elementary theory of formal power series rings. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 1–9. - Françoise Delon. Hensel fields in equal characteristic p &gt; 0. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 108–116. - Volume 50 Issue 3

Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Pade approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations.

In this chapter we prepare the ground for the proof of the Jung-Abhyankar theorem in section 2 and the study of quasiordinary power series in section 4 of chapter V. We assume that the reader is acquainted with the notion of power series over a field; in section 1, for the convenience of the reader, we give some background, introduce convergent power series over the real and complex numbers in section 2, and prove Weierstraß division and preparation theorem in section 3. The category of formal (resp. analytic) algebras—these are homomorphic images of formal (resp. convergent) rings of power series—is treated in section 4. The last section considers, in particular, the completion of an analytic algebra; these results are needed to prove the convergent case of the Jung-Abhyankar theorem.

An algorithm of multisummation of formal power series, solutions of linear ODE equations

Reduction of everywhere convergent power series with respect to Gröbner bases