scholarly journals Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations

1994 ◽  
Vol 1 (1) ◽  
pp. 1-13 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Asymptotic Series ◽  
Linear Differential Equations ◽  
Series Solutions ◽  
Linear Differential

scholarly journals Simplified error bounds for turning point expansions

2020 ◽  
pp. 1-32
Author(s):  
T. M. Dunster ◽  
A. Gil ◽  
J. Segura
Keyword(s):  
Differential Equations ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Linear Differential Equations ◽  
Airy Functions ◽  
Elementary Functions ◽  
Varying Coefficient ◽  
Linear Differential ◽  
High Degree

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, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver.

scholarly journals Space-time caustics

1986 ◽  
Vol 9 (3) ◽  
pp. 531-540 ◽  
Author(s):  
Arthur D. Gorman
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Series Solution ◽  
Asymptotic Series ◽  
Inhomogeneous Media ◽  
Linear Differential Equations ◽  
Spatially Inhomogeneous ◽  
Asymptotic Series Solution ◽  
Linear Differential

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of linear differential equations modelling wave propagation in spatially inhomogeneous media at caustic (turning) points. Here the formalism is adapted to determine a class of asymptotic solutions at caustic points for those equations modelling wave propagation in media with both spatial and temporal inhomogeneities. The analogous Schrodinger equation is also considered.

scholarly journals On the asymptotic series solution of some higher order linear differential equations at turning points

1984 ◽  
Vol 7 (3) ◽  
pp. 541-548
Author(s):  
Arthur D. Gorman
Keyword(s):  
Differential Equations ◽  
Series Solution ◽  
Turning Points ◽  
Asymptotic Series ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Asymptotic Series Solution ◽  
Linear Differential

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of second-order “wave type” differential equations at turning points. The formalism also applies to higher order linear differential equations, as we make explicit here illustrating with some4th order equations of physical significance.

scholarly journals Laplace Contour Integrals and Linear Differential Equations

2021 ◽  
Author(s):  
Norbert Steinmetz
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Distribution Of Zeros ◽  
Polynomial Solutions ◽  
Contour Integrals ◽  
Nevanlinna Functions ◽  
Linear Differential

AbstractThe purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ Λ ( z ) = 1 2 π i ∫ C ϕ ( t ) e - z t d t that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ L [ w ] ( z ) : = w ( n ) + ∑ j = 0 n - 1 ( a j + b j z ) w ( j ) = 0 . This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.

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