Boundary layer expansions for initial value problems with two complex time variables

Abstract

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter ǫ. We construct inner and outer solutions of the problem and relate them to asymptotic representations via Gevrey asymptotic expansions with respect to ǫ, in adequate domains. The construction of such analytic solutions is closely related to the procedure of summation with respect to an analytic germ, put forward in [19], whilst the asymptotic representation leans on the cohomological approach determined by Ramis-Sibuya Theorem.under given initial data u(0, t 2 , z, ǫ) ≡ u(t 1 , 0, z, ǫ) ≡ 0. Here, Q(X) ∈ C[X] and P (T 1 , T 2 , Z, z, ǫ) stands for a polynomial in (T 1 , T 2 , Z) with holomorphic coefficients w.r.t. (z, ǫ) on H β ×D(0, ǫ 0 ), where H β stands for the horizontal strip in the complex planefor some β > 0, and D(0, ǫ 0 ) ⊆ C stands for the open disc centered at the origin with radius ǫ 0 , for some small ǫ 0 > 0. The symbol ǫ acts as a small complex perturbation parameter in the *

On a q—Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

“…Such solutions are known as inner and outer solutions (see Definition 8 and Definition 9, resp.). Such asymptotic solutions have also been observed in the previous study [15], in the framework of singularly perturbed PDEs. However, the different nature of the asymptotic expansions regarding the outer and inner solutions is a novel phenomena which has firstly been observed in the present study.…”

“…under null initial data u(t 1 , 0, z, ) ≡ u(0, t 2 , z, ) ≡ 0, and where Q(X) ∈ C[X], and P is a polynomial with respect to its first three variables, with holomorphic coefficients on H β ×D(0, 0 ), and the forcing term is holomorphic on C × C × H β × (D(0, 0 ) \ {0}). The analytic solutions and their asymptotic expansions of those singularly perturbed partial differential equations are obtained in [15]. More precisely, the so-called inner solutions are holomorphic solutions of (2), holomorphic on domains in time which depend on the perturbation parameter and approach infinity, admits Gevrey asymptotic expansion of certain positive order, with respect to , whereas the so-called outer solutions are holomorphic solutions of (2), holomorphic on a product of finite sectors with vertex at the origin with respect to the time variables, admit Gevrey asymptotic expansion of a different positive order, with respect to .…”

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Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed q−difference-differential equations in the complex domain are constructed. They stand for a q−analog of the singularly perturbed partial differential equations considered in [15]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to the perturbation parameter.

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Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms

“…This, at first sight slight, variation on the form of the main problem varies its underlying geometry radically. On the other hand, the appearance of different types of solutions observed in [12], known as inner and outer solutions, which describe boundary layer expansions do not appear in the present situation, since we study local solutions in time t 1 , t 2 near the origin in the complex domain. It is worth mentioning that, despite the fact that the form of the main equation under study resembles that of [12], the nature of the singularities appearing in the problem require one to appeal different approaches and apply novel techniques, to be briefly described below.…”

“…The concrete assumptions on the elements involved in the main problem (1) are to be described and analyzed in detail throughout the work. The study of a problem of such form is motivated by the recent research [12] of the second and third authors. The main aim in the preceding work was related to the descrip-tion of the asymptotic behavior of the analytic solutions, with respect to the perturbation parameter, near the origin, of singularly perturbed equations…”