ASYMPTOTIC AND CONVERGENT EXPANSIONS FOR SOLUTIONS OF THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH A LARGE PARAMETER

作者：Ferreira Chelo; Lopez Jose L.^*; Perez Sinusia Ester

来源：Journal of Applied Analysis and Computation, 2018, 8(3): 965-981.

DOI：10.11948/2018.965

• Third-order differential equations
• asymptotic expansions
• Green's functions
• Banach's fixed point theorem
• Pearcey integral

In 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 ''' + a Lambda(2)y' + b Lambda(3)y = f(x)y' + g(x)y, with a, b is an element of C fixed, f' and g continuous, and Lambda 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 vertical bar x vertical bar.

• 出版日期2018-6