Singularly perturbed reaction diffusion equations with time delay

莫嘉琪,; 温朝晖,

doi:10.1007/s10483-010-1311-6

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…In the framework of partial differential equations, we mention [28] and [32] where formal asymptotic expansion like (6) have been obtained for solutions to reaction-diffusion equations with small delay.…”

Section: Resultsmentioning

confidence: 99%

On singularly perturbed small step size difference-differential nonlinear PDEs

Malek

2013

Journal of Difference Equations and Applications

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We study a family of singularly perturbed small step size difference-differential nonlinear equations in the complex domain. We construct formal solutions to these equations with respect to the perturbation parameter which are asymptotic expansions with 1−Gevrey order of actual holomorphic solutions on some sectors in near the origin in C. However, these formal solutions can be written as sums of formal series with a corresponding decomposition for the actual solutions which may possess a different Gevrey order called 1 + −Gevrey in the literature. This phenomenon of two levels asymptotics has been already observed in the framework of difference equations, see [6]. The proof rests on a new version of the so-called Ramis-Sibuya theorem which involves both 1−Gevrey and 1 + −Gevrey orders. Namely, using classical and truncated Borel-Laplace transforms (introduced by G. Immink in [18]), we construct a set of neighboring sectorial holomorphic solutions and functions whose difference have exponentially and super-exponentially small bounds in the perturbation parameter.

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Section: Resultsmentioning

confidence: 99%

On singularly perturbed small step size difference-differential nonlinear PDEs

Malek

2013

Journal of Difference Equations and Applications

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“…This is a field under strong research, namely for optimal control problems, differential equations, biology, etc (see e.g. [10,11,16,17,20,27,30,33]). For some literature on what this paper concerns, we suggest the reader to [2,4,6,8,9,12,18,19,23,31] for fractional variational problems dealing with Caputo derivative, in [3] for Lagrangians depending on fractional integrals, and in [13,21] when presence of indefinite integrals.…”

Section: Introductionmentioning

confidence: 99%