Date of Award
Master of Science (MS)
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 reformulating 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 other techniques, our solution maintains all the original function's information.
We discover a type of oscillation behavior hidden in the hyperasymptotic series. This behavior ought to be in the structure because of the way that the hyperseries is constructed. Each term in the hyperseries is in the form that consists of indexes and arguments. Each index changes discretely, whereas each argument changes continuously. This is the fundamental reason that causes the oscillation phenomenon. The other researchers have not addressed this issue in their related work because they only approximate the solution when the argument is at a single value.
Chen, Yan, "Asymptotic Series Solutions to One-Dimensional Helmholtz Equation" (2014). Applied Mathematics Graduate Theses & Dissertations. 54.