Abstract
A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.
Highlights
This work falls in the continuance of [1], where families of singularly perturbed initial value problems with the following shape
There, Q, Q1, Q2, R D stand for polynomials with complex coefficients and δD ≥ 2 is an integer
In the second central result of the paper, we show that the holomorphic inner solutions to (11)
Summary
This work falls in the continuance of [1], where families of singularly perturbed initial value problems with the following shape. For problems related to obstruction for analytic integrability of Hamiltonian systems and transseries expansions of first integrals, we refer to [8] Another aspect for which transseries turn out to be a powerful tool is the resurgence property of formal power series solutions to differential or more general functional equations (i.e., analytic continuation of their Borel transforms). The last section is devoted to the conclusion of the work where insights for prospective works are outlined
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