000056093 001__ 56093 000056093 005__ 20200221144339.0 000056093 0247_ $$2doi$$a10.1007/s00365-015-9298-y 000056093 0248_ $$2sideral$$a92267 000056093 037__ $$aART-2016-92267 000056093 041__ $$aeng 000056093 100__ $$0(orcid)0000-0002-3698-6719$$aFerreira, C.$$uUniversidad de Zaragoza 000056093 245__ $$aOn a Modification of Olver’s Method: A Special Case 000056093 260__ $$c2016 000056093 5060_ $$aAccess copy available to the general public$$fUnrestricted 000056093 5203_ $$aWe 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 (Formula presented.): (Formula presented.), with (Formula presented.) and g continuous. Olver studies in detail the cases (Formula presented.), especially the cases (Formula presented.), giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case (Formula presented.) is different, as the behavior of the solutions for large (Formula presented.) 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 (Formula presented.). 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. 000056093 536__ $$9info:eu-repo/grantAgreement/ES/MICINN/MTM2010-21037 000056093 540__ $$9info:eu-repo/semantics/openAccess$$aby-nc-nd$$uhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/ 000056093 590__ $$a0.964$$b2016 000056093 591__ $$aMATHEMATICS$$b67 / 310 = 0.216$$c2016$$dQ1$$eT1 000056093 592__ $$a1.094$$b2016 000056093 593__ $$aComputational Mathematics$$c2016$$dQ1 000056093 593__ $$aMathematics (miscellaneous)$$c2016$$dQ1 000056093 593__ $$aAnalysis$$c2016$$dQ2 000056093 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/acceptedVersion 000056093 700__ $$aLópez, J.L. 000056093 700__ $$0(orcid)0000-0002-8021-2745$$aPérez Sinusía, E.$$uUniversidad de Zaragoza 000056093 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada 000056093 773__ $$g43, 2 (2016), 273–290$$pConstr. approx.$$tCONSTRUCTIVE APPROXIMATION$$x0176-4276 000056093 8564_ $$s1108701$$uhttps://zaguan.unizar.es/record/56093/files/texto_completo.pdf$$yPostprint 000056093 8564_ $$s8429$$uhttps://zaguan.unizar.es/record/56093/files/texto_completo.jpg?subformat=icon$$xicon$$yPostprint 000056093 909CO $$ooai:zaguan.unizar.es:56093$$particulos$$pdriver 000056093 951__ $$a2020-02-21-13:48:58 000056093 980__ $$aARTICLE