Person: López García, José Luis
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López García
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José Luis
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Estadística, Informática y Matemáticas
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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas
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0000-0002-6050-9015
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2369
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Publication Open Access On a modifcation of Olver's method: a special case(Springer US, 2016) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika IngeniaritzaWe consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter : xm y −2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, especially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case m = 2. We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.Publication Open Access Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter(Shanghai Normal University, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika IngeniaritzaIn previous papers [6–8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver’s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: y ′′′ +aΛ2y′ +bΛ3y = f(x)y′ +g(x)y, with a, b ∈ C fixed, f′ and g continuous, and Λ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver’s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral P(x, y) for large |x|.