Journal of Integral Equations and Applications

Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III

Chelo Ferreira, José L. López, and Ester Pérez Sinusía

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We consider the asymptotic method designed by Olver \cite{olver} for linear differential equations of second order containing a large (asymptotic) parameter $\Lambda$, in particular, the second and third cases studied by Olver: differential equations with a turning point (second case) or a singular point (third case). It is well known that his method gives the Poincar\'e-type asymptotic expansion of two independent solutions of the equation in inverse powers of $\Lambda$. In this paper, we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then, using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence also has the property of being an asymptotic expansion for large $\Lambda$ (not of Poincar\'e-type) of the solution of the problem. Moreover, we show that the technique also works for nonlinear differential equations with a large parameter.

Article information

J. Integral Equations Applications, Volume 27, Number 1 (2015), 27-45.

First available in Project Euclid: 24 February 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 34B27: Green functions 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 45D05: Volterra integral equations [See also 34A12]

Second order differential equations turning points regular singular points Volterra integral equations of the second kind asymptotic expansions Green functions fixed point theorems airy functions Bessel functions


Ferreira, Chelo; López, José L.; Sinusía, Ester Pérez. Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III. J. Integral Equations Applications 27 (2015), no. 1, 27--45. doi:10.1216/JIE-2015-27-1-27.

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