On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

Abstract

A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

“…There is a rich and active study within the area of q-difference differential equations with dilations involving q > 1. These are highlighted by works of: L. Di Vizio [6][7][8]; C. Hardouin [7]; T. Dreyfus [9,10]; A. Lastra [10][11][12][13][14][15][16][17][18][19]; S. Malek [10][11][12][13][14][15][16][17][18][19][20][21][22]; J. Sanz [17][18][19]; H. Tahara [23]; and C. Zhang [8,24]; along with further references by these researchers and others. Often these studies in q-difference differential equations overlap with the area of Gevrey asymptotics.…”

“…For the given , set N = N 0 in (19) and (20). Then for |t| ≤ ρ and all 1 < q < min{q 0 , q 1 }, applying the bounds (21) and (22) to (20) gives…”

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

“…In addition to that, we assume that uniform bounds with respect to the perturbation parameter are satisfied, i.e. there exist C 1 2 > 0 with (20) sup…”

“…In this section, we preserve the values of the elements involved in the main problem (17) stated in Section 3. More precisely, we assume (11)- (16), and also the hypotheses on the forcing term (19) in (18) and the coefficients in (20). Let d ∈ R and S d an infinite sector with vertex at 0 ∈ C under the geometric condition imposed in Proposition 6.…”

“…We recall the definition of asymptotic expansion of mixed order as introduced in the work [20]. Observe that the set of functions admitting q−Gevrey asymptotic expansions of order 1/k coincide with the functions admitting ((q, k); 0)−Gevrey asymptotic expansion.…”