Exponential asymptotics: multi-level asymptotics of model problems - Say, Fatih (2016)
Exponential asymptotics, which deals with the interpretation of divergent series, is a highly topical field in mathematics. Exponentially small quantities frequently arise in applications, and Poincar´e’s definition of an asymptotic expansion, unfortunately, fails to emphasise the importance of such small exponentials, as they are hidden behind the algebraic order terms. In this thesis, we introduce a new method of hyperasymptotic expansion by inspecting resultant remainders of series. We study the method from two different concepts. First, deriving the singularities and the late order terms, where we truncate expansions at the least value and observe if the remainder is exponentially small. Substitution of the truncated remainder into original differential equation generates an inhomogeneous differential equation for the remainders. We expand the remainder as an asymptotic power series, and then the truncation leads to a new remainder which is exponentially smaller whence the related error estimate gets smaller, so that the numerical precision increases. Systematically repeating this process of reexpansions of the truncated remainders derives the exponential improvement in the approximate solution of the expansions and minimises the ignored terms, i.e., error estimate. Second, in establishing the level one error, which is a function of level zero and level one truncation points, we study asymptotic behaviour in terms of the truncation points and allow them to vary. Writing the estimate as a function of the preceding level truncation point and varying the number of the terms decreases the error dramatically. We also discuss the Stokes lines originating from the singularities of the expansion(s) and the switching on and off behaviour of the subdominant exponentials across these lines. A key result of this thesis is that when the higher levels of the expansions are considered in terms of the truncation points of preceding stages, the error estimate is minimised. This is demonstrated via several differential equations provided in the thesis.