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

Chelo Ferreira; José L. López; Ester Pérez Sinusía

doi:10.11948/2018.965

2018 Volume 8 Issue 3

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Chelo Ferreira, José L. López, Ester Pérez Sinusía. ASYMPTOTIC AND CONVERGENT EXPANSIONS FOR SOLUTIONS OF THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH A LARGE PARAMETER[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 965-981. doi: 10.11948/2018.965

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Chelo Ferreira, José L. López, Ester Pérez Sinusía. ASYMPTOTIC AND CONVERGENT EXPANSIONS FOR SOLUTIONS OF THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH A LARGE PARAMETER[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 965-981. doi: 10.11948/2018.965

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

1 Department of Applied Mathematics, University of Zaragoza and IUMA, Pedro Cerbuna, 12, 50012, Zaragoza, Spain;
2 Department of Mathematical and Computer Engineering, State University of Navarra and INAMAT, Campus de Arrosadia, s/n, 31006 Pamplona, Spain

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Abstract

Abstract

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''' + 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|.
- Third-order differential equations /
- asymptotic expansions /
- Green's functions /
- Banach's fixed point theorem /
- Pearcey integral
MSC: 34A12;41A58;41A60;34B27

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References

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