Differential equations defined by (convergent) Laurent series

2021 ◽  
Author(s):  
Vakhtang Lomadze
Keyword(s):  
Differential Equations ◽  
Laurent Series

scholarly journals Weakly perturbed boundary-value problems for systems of integro-differential equations with impulsive action

2015 ◽  
Vol 63 (1) ◽  
pp. 73-87
Author(s):  
Ivanna Bondar
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Boundary Value Problems ◽  
Laurent Series ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Impulsive Action ◽  
Differential Systems

Abstract The weakly perturbed BVP's for impulsive integro-differential systems are considered. Under the assumption that the generating problem (for ε = 0) does not have solutions on the space W12[a,b] for some inhomogeneity and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these problems on the space D2([a,b]{τi}I) in the form of a Laurent series in powers of small parameter ε with finitely many terms with negative powers of ε, and we suggest an algorithm of construction of these solutions.

8.—On a Class of Series Expansions in the Theory of Emden's Equation

1973 ◽  
Vol 71 (2) ◽  
pp. 95-110 ◽  
Author(s):  
Einar Hille
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Series Expansion ◽  
Laurent Series ◽  
Second Order ◽  
Series Expansions ◽  
Section 8 ◽  
Second Order Differential Equations ◽  
Bona Fide ◽  
Image Position

SynopsisThis paper deals with the nature of movable singularities of solutions of Emden's equationat which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c asBy the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the formwhere Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

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