°µÍø½ûÇø

Asymptotic series solutions to the one-dimensional Helmholtz equations

Yan Chen

Applied Mathematics,Ìý

Date and time:Ìý

Tuesday, April 15, 2014 - 10:00am

³¢´Ç³¦²¹³Ù¾±´Ç²Ô:Ìý

ECOT 314

´¡²ú²õ³Ù°ù²¹³¦³Ù:Ìý

°Õ³ó±ðÌýNthÌýpartial sum of an infinite power series has the form Σ±·²Ô=1Ìý³¦²Ô³æ²Ô, where x is a parameter and {cn} are known numbers. The infinite series isÌýconvergentÌýif this sum has a finite limit as N → ∞, with x fixed. The series isÌýasymptoticÌýif the error incurred by truncating the series after N terms is O(x(N+1)) as x → 0. The two concepts, which involve two different limits, are independent - a given series might have both properties, or one, or neither.

Divergent asymptotic series can be quite useful: the well known formula of Stirling (1730) [?] for ln(m!) for large m is a divergent series (in powers of 1/m) for all finite m, but Stirling calculated log10(1000!) to ten decimal places using only a few terms in his series. (see Copson, (1967) [10]).

Divergent asymptotic series are most useful when the parameter x is small but not zero. In this situation, with x fixed, one finds that the error gets smaller as one keeps more terms in the series until a specific point, after which the error begins to grow. In this way, a divergent asymptotic series can approximate a function for small, fixed x up to some optimal accuracy, but no better. Dingle (1973) [?] began development of a method to create a follow-on series, which begins at the point of optimal accuracy of the original series, and can be used to improve the accuracy further, but again with a minimal error that cannot be reduced by further use of the two series. Berry & Howls (1990) [?] developed Dingle’s idea further, using a sequence of divergent series, with each series improving on the accuracy of the preceding one. Again, they eventually came to a (much smaller) minimal error that they could not reduce. They named this sequence of increasingly accurate asymptotic seriesÌýhyperasymptotics. They demonstrated their approach by developing hyperasymptotic series, valid for large positive z, for the Airy function, Ai(z), and showed that they could evaluate Ai(z) with 20 decimal places of accuracy, even down toÌýzÌý= 5:241.

We present a variation of the method of Berry & Howls, which eliminates some of the inherent error in their approach. Using Dingle’s change of variables we transform the Airy differential equation into a new ODE that is exact, for all positive z. After reforming the ODE as an integral equation, we solve the integral equation (exactly) with a recurrent series that converges absolutely for all positive z. Each term in our series can be expanded as an asymptotic series with the error term under our control because of the bound we develop for it. Comparing with Berry’s hyperasymptotic series technique, our solution maintains all the original function’s information that Berry’s method has lost.

We also discover a type of oscillation behavior hidden in the hyperasymptotic series. As far as we know, this issue has not been addressed in other literature before. This behavior ought to be hidden in the structure because of the way people construct the series. Each term in the series is in the form that consists of the index and the arguments. The change of index is discretely, while as the arguments change continously. This is the fundamental reason that causes the oscillation phenomenon. Our hypothesis is that other researchers, like Berry, Howls and Boyd, did not discover this issue in their related literature is because the approximation methods, such as Borel summation and Euler acceleration method, have eliminated this information.