Singular Perturbation of Nonlinear Systems with Regular Singularity
We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal...
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| Format: | Article |
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Wiley
2018-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2018/5163492 |
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| author | Domingos H. U. Marchetti William R. P. Conti |
| author_facet | Domingos H. U. Marchetti William R. P. Conti |
| author_sort | Domingos H. U. Marchetti |
| collection | DOAJ |
| description | We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type. |
| format | Article |
| id | doaj-art-7821b856aaa54b6783080db07e000282 |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-7821b856aaa54b6783080db07e0002822025-08-20T02:07:08ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2018-01-01201810.1155/2018/51634925163492Singular Perturbation of Nonlinear Systems with Regular SingularityDomingos H. U. Marchetti0William R. P. Conti1Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, SP, BrazilDepartamento de Ciências do Mar, Universidade Federal de São Paulo, Rua Dr. Carvalho de Mendonça 144, 11070-100 Santos, SP, BrazilWe extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.http://dx.doi.org/10.1155/2018/5163492 |
| spellingShingle | Domingos H. U. Marchetti William R. P. Conti Singular Perturbation of Nonlinear Systems with Regular Singularity Discrete Dynamics in Nature and Society |
| title | Singular Perturbation of Nonlinear Systems with Regular Singularity |
| title_full | Singular Perturbation of Nonlinear Systems with Regular Singularity |
| title_fullStr | Singular Perturbation of Nonlinear Systems with Regular Singularity |
| title_full_unstemmed | Singular Perturbation of Nonlinear Systems with Regular Singularity |
| title_short | Singular Perturbation of Nonlinear Systems with Regular Singularity |
| title_sort | singular perturbation of nonlinear systems with regular singularity |
| url | http://dx.doi.org/10.1155/2018/5163492 |
| work_keys_str_mv | AT domingoshumarchetti singularperturbationofnonlinearsystemswithregularsingularity AT williamrpconti singularperturbationofnonlinearsystemswithregularsingularity |