On Some Parabolic Type Problems from Thin Film Theory and Chemical Reaction-Diffusion Networks

Abstract

principle (the continuous and the discrete versions; see chapter 6) imposes a limitation on the spatial dimension (should be at most five; see [24] for details). The Method of Lines (MOL) is not a mainstream numerical tool and the specialized literature is rather scarce. The method amounts to discretizing evolutionary PDE's in space only, so it produces a semi-discrete numerical scheme which consists of a system of ODE's (in the time variable). To prove convergence of the semi-discrete MOL scheme to the original PDE one needs to perform some more or less traditional analysis: it is necessary to show that the scheme is consistent with the continuous problem, and that the discretized version of the spatial differential operator retains sufficient dissipative properties in order to allow an application of Gronwall's Lemma to the error term. As shown in [23], a uniform (in time) consistency estimate is sufficient to obtain convergence; however, the consistency estimate we proved is not uniform for small time, so we cannot directly employ the results in [23] to prove convergence in our case. Instead, we prove all the required estimates "from the scratch", then we use their exact quantitative form in order to conclude convergence. iv DEDICATION I would like to dedicate this work to My parents, my husband, my children, my siblings. v ACKNOWLEDGEMENTS First and foremost, I am most indebted to my advisor Dr. Adrian Tudorascu and my co-advisor Dr. Casian Pantea for their continued encouragement and support over these last few years. It has been a pleasure to work under their supervision. Without them, this dissertation could not have come about. Also, I would like to thank Dr.Pantea for providing me with financial support as a research assistant: Part I is joint work with Dr.