93
Views
9
CrossRef citations to date
0
Altmetric
Article

On singularly perturbed small step size difference-differential nonlinear PDEs

Pages 118-168 | Received 06 Feb 2013, Accepted 06 Jun 2013, Published online: 18 Jul 2013

References

  • Z.Artstein and M.Slemrod, On singularly perturbed retarded functional-differential equations, J. Differ. Equ.171(1) (2001), pp. 88–109.
  • W.Balser, From Divergent Power Series to Analytic Functions, Theory and Application of Multisummable Power Series. Lecture Notes in Mathematics, Vol. 1582, Springer-Verlag, Berlin, 1994.
  • B.Braaksma and B.Faber, Multisummability for some classes of difference equations, Ann. Inst. Fourier (Grenoble)46(1) (1996), pp. 183–217.
  • B.Braaksma, B.Faber, and G.Immink, Summation of formal solutions of a class of linear difference equations, Pacific J. Math.195(1) (2000), pp. 35–65.
  • O.Costin, Asymptotics and Borel summability, Chapman Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 141, CRC Press, Boca Raton, FL, 2009.
  • O.Costin and S.Tanveer, Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Comm. Pure Appl. Math.53(9) (2000), pp. 1092–1117.
  • J.Dieudonné, Calcul infinitésimal (French), Hermann, Paris 1968, p. 479.
  • A.El-Rabih and R.Schäfke, Overstable analytic solutions for non-linear systems of difference equations with small step size containing an additional parameter, J. Differ. Equ. Appl.11(3) (2005), pp. 183–213.
  • A.Fowler, Asymptotic methods for delay equations, J. Engrg. Math.53(3–4) (2005), pp. 271–290.
  • A.Fruchard and R.Schäfke, Bifurcation delay and difference equations, Nonlinearity16(6) (2003), pp. 2199–2220.
  • A.Fruchard and R.Schäfke, Analytic solutions of difference equations with small step size. In memory of W.A. Harris, J. Differ. Equ. Appl.7(5) (2001), pp. 651–684.
  • V.Glizer, Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution, J. Math. Anal. Appl.278(2) (2003), pp. 409–433.
  • P.Hsieh and Y.Sibuya, Basic Theory of Ordinary Differential Equations, Universitext. Springer-Verlag, New York, 1999.
  • G.Immink, Accelero-summation of the formal solutions of nonlinear difference equations, Ann. Inst. Fourier (Grenoble)61(1) (2011), pp. 1–51.
  • G.Immink, Exact asymptotics of nonlinear difference equations with levels 1 and 1+, Ann. Fac. Sci. Toulouse Math.17(2) (2008), pp. 309–356.
  • G.Immink, Asymptotic Expansions with Error Bounds for Solutions of Difference Equations of ‘Level 1+’. Équations différentielles dans le champ complexe, Vol. I (Strasbourg, 1985), Publ. Inst. Rech. Math. Av., Univ. Louis Pasteur, Strasbourg, 1988.
  • A. Lastra and S. Malek, On q-Gevrey asymptotics for singularly perturbed q-difference-differential problems with an irregular singularity, Abstract and Applied Analysis, Vol. 2012, Article ID 860716.
  • S. Malek, On the parametric Stokes phenomenon for solutions of singularly perturbed linear partial differential equations, Abstract and Applied Analysis, Vol. 2012, Article ID 930385.
  • S.Malek, On the summability of formal solutions for doubly singular nonlinear partial differential equations, J. Dynam. Control. Syst.18(1) (2012), pp. 45–82.
  • S.Malek, Singularly perturbed q-difference-differential equations with irregular singularity, J. Dynam. Control. Syst.17(2) (2011), pp. 243–271.
  • B.Malgrange and J.-P.Ramis, Fonctions multisommables, Ann. Inst. Fourier, Grenoble42(1) (1992), pp. 353–368.
  • J.Melenk and C.Schwab, Analytic regularity for a singularly perturbed problem, SIAM J. Math. Anal.30(2) (1999), pp. 379–400.
  • Jia-qiMo and Zhao-huiWen, Singularly perturbed reaction diffusion equations with time delay, Appl. Math. Mech. (English Ed.)31(6) (2010), pp. 769–774.
  • F.Olver, Asymptotics and Special Functions. Computer Science and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, xvi+572pp, 1974.
  • J.-P.Ramis and Y.Sibuya, A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble)44(3) (1994), pp. 811–848.
  • H.Tian, Asymptotic expansion for the solution of singularly perturbed delay differential equations, J. Math. Anal. Appl.281(2) (2003), pp. 678–696.
  • Z.Xiang, Singular perturbation of initial-boundary value problems of reaction diffusion equation with delay, Appl. Math. Mech. (English Ed.)15(3) (1994), pp. 267–273.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.