On Complex Singularity Analysis for Some Linear Partial Differential Equations in
We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/394564 |
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| author | A. Lastra S. Malek C. Stenger |
| author_facet | A. Lastra S. Malek C. Stenger |
| author_sort | A. Lastra |
| collection | DOAJ |
| description | We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series. |
| format | Article |
| id | doaj-art-3dd90a7104c542fea2329f507d3a0b49 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-3dd90a7104c542fea2329f507d3a0b492025-08-20T03:55:40ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/394564394564On Complex Singularity Analysis for Some Linear Partial Differential Equations inA. Lastra0S. Malek1C. Stenger2Facultad de Ciencias, University of Alcalá, Apartado de Correos 20, 28871 Alcalá de Henares (Madrid), SpainUFR de Mathématiques, University of Lille 1, Cité Scientifique M2, 59655 Villeneuve d'Ascq Cedex, FranceLaboratoire Mathématiques, Images et Applications, University of La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle Cedex, FranceWe investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.http://dx.doi.org/10.1155/2013/394564 |
| spellingShingle | A. Lastra S. Malek C. Stenger On Complex Singularity Analysis for Some Linear Partial Differential Equations in Abstract and Applied Analysis |
| title | On Complex Singularity Analysis for Some Linear Partial Differential Equations in |
| title_full | On Complex Singularity Analysis for Some Linear Partial Differential Equations in |
| title_fullStr | On Complex Singularity Analysis for Some Linear Partial Differential Equations in |
| title_full_unstemmed | On Complex Singularity Analysis for Some Linear Partial Differential Equations in |
| title_short | On Complex Singularity Analysis for Some Linear Partial Differential Equations in |
| title_sort | on complex singularity analysis for some linear partial differential equations in |
| url | http://dx.doi.org/10.1155/2013/394564 |
| work_keys_str_mv | AT alastra oncomplexsingularityanalysisforsomelinearpartialdifferentialequationsin AT smalek oncomplexsingularityanalysisforsomelinearpartialdifferentialequationsin AT cstenger oncomplexsingularityanalysisforsomelinearpartialdifferentialequationsin |