Singularly perturbed problems in partial differential equations: a survey

Kadalbajoo, Mohan K.; Patidar, Kailash C.

doi:10.1016/s0096-3003(01)00291-0

Cited by 45 publications

(23 citation statements)

References 123 publications

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“…Equation (3) falls in the framework of the asymptotic study of singularly perturbed problems which have been studied in the case of a real perturbation parameter and P 1 acts on C ∞ (R d ) functions or Sovolev spaces H(R d ). We refer to [6] for the details. The case of a complex perturbation parameter has led to results in which the nature of the singularities arising from the problem describe different types of singularities.…”

Section: Introductionmentioning

confidence: 99%

On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Lastra

Malek

2017

Journal of Difference Equations and Applications

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We study the asymptotic behavior of the solutions related to a singularly perturbed q-differencedifferential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and q−Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution.The proof leans on a two level novel version of Ramis-Sibuya theorem under Gevrey and q-Gevrey orders.

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Section: Introductionmentioning

confidence: 99%

On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Lastra

Malek

2017

Journal of Difference Equations and Applications

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“…The proofs use methods of the theory of semigroups of operators (see [17]), the maximum principle and estimates of the energy integrals (see [15,25]), or fixed-point theorems for nonlinear equations (see [13,15]). For a general survey on singular perturbations for both asymptotic and numerical aspects, we refer the author to [19]. But, there are very few information on singularly perturbed partial differential equations with complex parameter and with solutions in spaces of analytic functions.…”

Section: S Malekmentioning

confidence: 99%

On the summability of formal solutions for doubly singular nonlinear partial differential equations

Malek

2012

J Dyn Control Syst

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We study Gevrey asymptotic properties of solutions of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain. We construct actual holomorphic solutions of these problems with the help of the BorelLaplace transforms. Using the Malgrange-Sibuya theorem, we show that these holomorphic solutions have a common formal power series asymptotic expansion of Gevrey order 1 in the perturbation parameter.

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“…It is a powerful instrument for analysis of singular perturbation problems. It has been studied for many types of differential equations; partial differential equations [7], singularly perturbed differential inclusions [8], functional-differential inclusions [9], discontinuous differential equations [10,11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%