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Gevrey Asymptotics for Logarithmic‐Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

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  • Stéphane Malek

Abstract

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter ϵ known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic‐type functions in time and are bounded holomorphic in space. A set of logarithmic‐type solutions are shaped by means of Laplace transforms relatively to t and ϵ and Fourier integrals in space. Furthermore, a formal logarithmic‐type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to ϵ on bounded sectors at the origin.

Suggested Citation

  • Stéphane Malek, 2023. "Gevrey Asymptotics for Logarithmic‐Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities," Abstract and Applied Analysis, John Wiley & Sons, vol. 2023(1).
    • Handle: RePEc:wly:jnlaaa:v:2023:y:2023:i:1:n:3025513
    DOI: 10.1155/2023/3025513
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    References listed on IDEAS

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    1. Stéphane Malek & Ying Hu, 2023. "Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities," Abstract and Applied Analysis, Hindawi, vol. 2023, pages 1-42, April.
    2. P. Hammachukiattikul & E. Sekar & A. Tamilselvan & R. Vadivel & N. Gunasekaran & Praveen Agarwal & Xiaolong Qin, 2021. "Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations," Journal of Mathematics, Hindawi, vol. 2021, pages 1-15, June.
    3. Avudai Selvi, P. & Ramanujam, N., 2017. "A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 101-115.
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