# Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations

@article{Dunster2020UniformAE,
title={Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations},
author={T. Mark Dunster},
journal={arXiv: Classical Analysis and ODEs},
year={2020}
}
• T. M. Dunster
• Published 12 October 2020
• Mathematics
• arXiv: Classical Analysis and ODEs
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions involve exponential, Airy and Scorer functions and slowly varying analytic coefficient functions involving simple coefficients. The approximations are uniformly valid for large values of the parameter and unbounded real and complex values of the argument…
2 Citations

## Figures from this paper

Nield-Kuznetsov Functions and Laplace Transforms of Parabolic Cylinder Functions
• T. M. Dunster
• Computer Science, Mathematics
SIAM J. Math. Anal.
• 2021
Nield-Kuznetsov functions of the first kind are studied, which are solutions of an inhomogeneous Weber parabolic cylinder differential equation, and have applications in fluid flow problems.
Recent Advances in the Nield-Kuznetsov Functions
• WSEAS TRANSACTIONS ON SYSTEMS
• 2021
In this article, we discuss a class of functions known as the Nield-Kuznetsov functions, introduced over the past decade. These functions arise in the solutions to inhomogeneous Airy’s and Weber’s

## References

SHOWING 1-10 OF 38 REFERENCES
Numerical and asymptotic aspects of parabolic cylinder functions
Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point
Computation of Asymptotic Expansions of Turning Point Problems via Cauchy’s Integral Formula: Bessel Functions
• Mathematics
• 2016
Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function
Integral representations for computing real parabolic cylinder functions
• Computer Science, Mathematics
Numerische Mathematik
• 2004
The new integrals follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the functions U,x, V, V and W and their derivatives by quadrature rules.
Simplified error bounds for turning point expansions
• Mathematics
• 2019
Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions,
Computing the real parabolic cylinder functions U(a, x), V(a, x)
• Computer Science, Mathematics
TOMS
• 2006
In an accompanying article, the precise domains for these methods are described and the Fortran 90 codes for the computation of these functions are presented.
Fast and accurate computation of the Weber parabolic cylinder function W(a, x)
• Mathematics
• 2011
textabstractMethods for the numerical evaluation of the Weber parabolic cylinder functions $W(a,\pm x)$, which are independent solutions of the inverted harmonic oscillator $y''+(x^2/4-a)y=0$, are
Asymptotic Methods For Integrals
This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods,
Liouville-Green expansions of exponential form, with an application to modified Bessel functions
• T. M. Dunster
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2020
Abstract Linear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z)
Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders
a re sought for la rge values of IILI, which a re uniformly valid wi th respect Lo a rg IL and un restricted va lues of the complex variable t. Two types of expa nsion are found . Those of the first