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The theme running throughout this thesis is the Painlevé equations, in their differential, discrete and ultra-discrete versions. The differential Painlevé equations have the Painlevé property. If all solutions of a differential equation are meromorphic functions then it necessarily has the Painlevé property. Any ODE with the Painlevé property is necessarily a reduction of an integrable PDE. Nevanlinna theory studies the value distribution and characterizes the growth of meromorphic functions, by using certain averaged properties on a disc of variable radius. We shall be interested in its well-known use as a tool for detecting integrability in difference equations—a difference equation may be integrable if it has sufficiently many finite-order solutions in the sense of Nevanlinna theory. This does not provide a sufficient test for integrability; additionally it must satisfy the well-known singularity confinement test.

The numerical approximation of solutions of ordinary differential equations played an important role in Numerical Analysis and still continues to be an active field of research. This is mainly due to the pressure of needs to model mathematically real world phenomena. In this thesis we are mainly concerned with the numerical solution of the first-order system of nonlinear two-point boundary value problems dy dx = f (x, y), a≤ x ≤ b, g(y(a), y(b)) = 0, where y ∈ Rn, f : R × Rn → Rn, and g : Rn × Rn → Rn. We will focus on the problem of solving singular perturbation problems since this has for many years been a source of difficulty to applied mathematicians and numerical analysts alike. We consider first deferred correction schemes based on Mono-Implicit Runge- Kutta (MIRK) and Lobatto formulae. As is to be expected, the scheme based on Lobatto formulae, which are implicit, is more stable than the scheme based on MIRK formulae which are explicit. Another deferred correction scheme, which uses the idea of the superconvergent deferred correction schemes, is also derived, and is shown to be highly stable compared to MIRK deferred correction schemes. To provide the continuous extension of the discrete solution, we construct high order interpolants based on an approach of using the already computed discrete solutions obtained on the final mesh. We will consider the construction of both explicit and implicit interpolants. An interpolation using a quasi-uniform grid is also introduced. This grid is naturally obtained in the mesh doubling which is a part of Richardson extrapolation. The estimation of conditioning numbers is discussed and used to develop mesh selection algorithms which will be appropriate for solving stiff linear and nonlinear boundary value problems. The algorithms are implemented in codes using deferred correction schemes based on MIRK and Lobatto formulae and the performance of codes which take account of the conditioning is compared with the performance of codes which use accuracy alone. Most of problems discussed in this thesis are two-point boundary value problems with separated boundary conditions. To complete our discussion, we explain numerical methods for solving two-point boundary value problems with nonseparated conditions, problems which contain parameters and those where the boundary conditions are given as integral constraints. We implement QR decomposition based on Householder transformations in the numerical experiments and discuss the results compared with Gaussian elimination.

This thesis concerns the development of a method for the detection of small solutions to delay differential equations. The detection of small solutions is important because their presence has significant influence on the analytical prop¬erties of an equation. However, to date, analytical methods are of only limited practical use. Therefore this thesis focuses on the development of a reliable new method, based on finite order approximations of the underlying infinite dimen¬sional problem, which can detect small solutions. Decisions (concerning the existence, or otherwise, of small solutions) based on our visualisation technique require an understanding of the underlying methodol¬ogy behind our approach. Removing this need would be attractive. The method we have developed can be automated, and at the end of the thesis we present a prototype Matlab code for the automatic detection of small solutions to delay differential equations.

Αποδείξεις θεωρημάτων για την τάξη των λύσεων των εξισώσεων ρητού τύπου και πολυωνυμικού τύπου με την βοήθεια της Θεωρίας NENVALINNA / -

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Dans cette thèse on s'intéresse à l'étude mathématique et numérique de modèles à frontières libres intervenant en physique et en biologie. Dans une première partie on considère un modèle de carbonatation des bétons armés. Ce modèle unidimensionnel est composé d'un système d'équations paraboliques de type réaction-diffusion défini sur un domaine où une interface est fixe et l'autre mobile. Cette interface mobile est solution d'une équation différentielle ordinaire et évolue au cours du temps suivant une loi en racine de t. Dans un premier temps, on définit pour ce modèle un schéma numérique de type volumes finis implicite/explicite en temps et on prouve la convergence de ce schéma. Dans un second temps, on construit un schéma volumes finis complètement implicite permettant de démontrer la propagation en racine de t de l'interface mobile au niveau discret. On s'intéresse ensuite à un système de diffusion croisée modélisant la croissance de biofilms. On introduit un schéma numérique de type volumes finis préservant la structure de flot de gradient du modèle. On prouve alors l'existence de solutions et la convergence du schéma. Enfin, on établit via des outils du transport optimal et du calcul des variations un résultat d'existence pour un modèle jouet de corrosion à frontière libre. Nous essayons par l'introduction de ce problème de mieux comprendre la structure du modèle DPCM (Diffusion-Poisson-Coupled-Model), également défini sur domaine mobile, décrivant la corrosion d'un baril métallique placé dans un milieu argileux (conditions de stockage des déchets nucléaires) et pour lequel il n'existe aucun résultat d'existence. / This thesis deals with the numerical and mathematical study of models with free boundaries coming from physics and biology. In the first part, we consider a model which describes the carbonnation phenomena in reinforced concrete. The model involves a system of 1D-parabolic equation of reaction diffusion type defined on a domain with a moving boundary. The motion of this interface is governed by an ordinary differential equation and it increases asymptotically as a square root of t for large times. We first introduce a Finite Volume numerical scheme for the model with implicit/explicit time discretization and we prove its convergence. Next, we build a fully implicit scheme for which we are able to establish the behavior in square root of t of the interface in this discrete setting. In a second part, we study a cross-diffusion system modeling the expansion of some biofilms. We introduce a numerical scheme of Finite Volumes type which preserves the gradient flow structure of the model. We establish the existence of solutions to the scheme and its convergence towards a solution to the original model. Eventually, we consider a toy model derived from a more complete model called DPCM (Diffusion-Poisson-Coupled-Model). The later describes the corrosion of (nuclear waste) containers made of iron and stored in clay soil. Again the model involves a free boundary whose position is part of the unknowns. Using tools from Optimal Transport Theory and Calculus of Variations, we establish the existence of a solution to the model. This is a first step towards the study of DPCM for which no such result is availiable.

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