Studies On Summability Of Formal Solution To A Cauchy Problem And On Integral Functions Of Mordell's Type

From formal solutions of ordinary or partial differential equations, one may give different sums by different summation processes. This phenomenon occurs for functional equations such as difference or q-difference equations. In this thesis, we shall consider the heat equation with a singular initial condition The aim is to give three sums of a divergent formal solution to this Cauchy problem:Borel-sum based on known results in [26] and two q-Borel sums obtained by means of Heat kernel and Jacobi theta function respectively (cf. [50] and [42,51]) and establish relations among them. More specifically, this thesis consists of the following six chapters.In Chapter 1, we introduce some known results on summability of formal solutions, state our problem and main conclusions, and recall how to solve Cauchy problem for the real Heat equation with Heat kernel.In Chapter 2, we introduce the classic Borel-Laplace summation and show the theorem on the finely Borel sum of divergent solutions of the com-plex Heat equation by Lutz, Miyake and Schafke (cf. [26]), and obtain the finely Borel sum of the formal solution to our problem.In Chapter 3, we introduce the so-called Gq-summation (cf. [50]). By variable substitutions, we can transfer the divergent formal solution to our Cauchy problem into a q-series. Then we obtain a q-Borel sum based on Heat kernel and compare the sum function defined in the previous sections.In Chapter 4, we firstly prove some properties of the Jacobi theta func-tion and introduce a method of summation based on Jacobi theta function (cf. [51]), and then get the other q-Borel sum of the q-series. In Chapter 5, we study integral functions which have been considered by Riemann, Kronecker, Lerch, Hardy, Ramanujan, Mordell and many other mathematicians. We say that Mordell's theorem (cf. [34,35]) about these integrals can be deduced from one of our other main theorems. And we can apply our ideas mentioned above to the more general cases.In Chapter 6, we sum up in a few sentences and provide some unsolved problems.