Unfolded singularities of analytic differential equations

La thèse est composée d’un chapitre de préliminaires et de deux articles sur le sujet du déploiement de singularités d’équations différentielles ordinaires analytiques dans le plan complexe. L’article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity traite le problème de l’équivalence analytique de familles paramétriques de systèmes linéaires en dimension 2 qui déploient une singularité résonante générique de rang de Poincaré 1 dont la matrice principale est composée d’un seul bloc de Jordan. La question: quand deux telles familles sontelles équivalentes au moyen d’un changement analytique de coordonnées au voisinage d’une singularité? est complètement résolue et l’espace des modules des classes d’équivalence analytiques est décrit en termes d’un ensemble d’invariants formels et d’un invariant analytique, obtenu à partir de la trace de la monodromie. Des déploiements universels sont donnés pour toutes ces singularités. Dans l’article Confluence of singularities of non-linear differential equations via Borel–Laplace transformations on cherche des solutions bornées de systèmes paramétriques des équations non-linéaires de la variété centre de dimension 1 d’une singularité col-noeud déployée dans une famille de champs vectoriels complexes. En général, un système d’ÉDO analytiques avec une singularité double possède une unique solution formelle divergente au voisinage de la singularité, à laquelle on peut associer des vraies solutions sur certains secteurs dans le plan complexe en utilisant les transformations de Borel–Laplace. L’article montre comment généraliser cette méthode et déployer les solutions sectorielles. On construit des solutions de systèmes paramétriques, avec deux singularités régulières déployant une singularité irrégulière double, qui sont bornées sur des domaines «spirals» attachés aux deux points singuliers, et qui, à la limite, convergent vers une paire de solutions sectorielles couvrant un voisinage de la singularité confluente. La méthode apporte une description unifiée pour toutes les valeurs du paramètre.

Invariant discretizations of partial differential equations

Un algorithme permettant de discrétiser les équations aux dérivées partielles (EDP) tout en préservant leurs symétries de Lie est élaboré. Ceci est rendu possible grâce à l'utilisation de dérivées partielles discrètes se transformant comme les dérivées partielles continues sous l'action de groupes de Lie locaux. Dans les applications, beaucoup d'EDP sont invariantes sous l'action de transformations ponctuelles de Lie de dimension infinie qui font partie de ce que l'on désigne comme des pseudo-groupes de Lie. Afin d'étendre la méthode de discrétisation préservant les symétries à ces équations, une discrétisation des pseudo-groupes est proposée. Cette discrétisation a pour effet de transformer les symétries ponctuelles en symétries généralisées dans l'espace discret. Des schémas invariants sont ensuite créés pour un certain nombre d'EDP. Dans tous les cas, des tests numériques montrent que les schémas invariants approximent mieux leur équivalent continu que les différences finies standard.

Analytical Equations for Compact Dual Frequency Microstrip Antenna

Design equations are presented for calculating the resonance frequencies for a compact dual frequency arrow-shaped microstrip antenna. This provides a fast and simple way to predict the resonant frequencies of the antenna. The antenna is also analyzed using the IE3D simulation package. The theoretical predictions are found to be very close to the IE3D results and thus establish the validity of the design formulae

Enhancement of Microwave Properties of Planar Filters and Antennas using Photonic Bandgap (PBG) Structures

In this thesis, we explore the design, computation, and experimental analysis of photonic crystals, with a special emphasis on structures and devices that make a connection with practically realizable systems. First, we analyze the propenies of photonic-crystal: periodic dielectric structures that have a band gap for propagation. The band gap of periodically loaded air column on a dielectric substrate is computed using Eigen solvers in a plane wave basis. Then this idea is extended to planar filters and antennas at microwave regime. The main objectives covered in this thesis are:• Computation of Band Gap origin in Photonic crystal with the abet of Maxwell's equation and Bloch-Floquet's theorem • Extension of Band Gap to Planar structures at microwave regime • Predict the dielectric constant - synthesized dieletric cmstant of the substrates when loaded with Photonic Band Gap (PBG) structures in a microstrip transmission line • Identify the resonant characteristic of the PBG cell and extract the equivalent circuit based on PBG cell and substrate parameters for microstrip transmission line • Miniaturize PBG as Defected Ground Structures (DGS) and use the property to be implemented in planar filters with microstrip transmission line • Extended the band stop effect of PBG / DGS to coplanar waveguide and asymmetric coplanar waveguide. • Formulate design equations for the PBG / DGS filters • Use these PBG / DGS ground plane as ground plane of microstrip antennas • Analysis of filters and antennas using FDID method

Fluid mechanics-non-linear waves studies on korteweg-de vries equations

In this thesis the author has presented qualitative studies of certain Kdv equations with variable coefficients. The well-known KdV equation is a model for waves propagating on the surface of shallow water of constant depth. This model is considered as fitting into waves reaching the shore. Renewed attempts have led to the derivation of KdV type equations in which the coefficients are not constants. Johnson's equation is one such equation. The researcher has used this model to study the interaction of waves. It has been found that three-wave interaction is possible, there is transfer of energy between the waves and the energy is not conserved during interaction.

Studies on some nonlinear schrodinger type equations describing pulse propagation through optical fibers

The discovery of the soliton is considered to be one of the most significant events of the twentieth century. The term soliton refers to special kinds of waves that can propagate undistorted over long distances and remain unaffected even after collision with each other. Solitons have been studied extensively in many fields of physics. In the context of optical fibers, solitons are not only of fundamental interest but also have potential applications in the field of optical fiber communications. This thesis is devoted to the theoretical study of soliton pulse propagation through single mode optical fibers.

Studies on integrability of some perturbed nonlinear evolution equations

Usually typical dynamical systems are non integrable. But few systems of practical interest are integrable. The soliton concept is a sophisticated mathematical construct based on the integrability of a class ol' nonlinear differential equations. An important feature in the clevelopment. of the theory of solitons and of complete integrability has been the interplay between mathematics and physics. Every integrable system has a lo11g list of special properties that hold for integrable equations and only for them. Actually there is no specific definition for integrability that is suitable for all cases. .There exist several integrable partial clillerential equations( pdes) which can be derived using physically meaningful asymptotic teclmiques from a very large class of pdes. It has been established that many 110nlinear wa.ve equations have solutions of the soliton type and the theory of solitons has found applications in many areas of science. Among these, well-known equations are Korteweg de-Vries(KdV), modified KclV, Nonlinear Schr6dinger(NLS), sine Gordon(SG) etc..These are completely integrable systems. Since a small change in the governing nonlinear prle may cause the destruction of the integrability of the system, it is interesting to study the effect of small perturbations in these equations. This is the motivation of the present work.

Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity

Artificial boundary conditions are presented to approximate solutions to Stokes- and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v^infinity, p^infinity to the problems in the unbounded domain Omega the error v^infinity - v^R, p^infinity - p^R is estimated in H^1(Omega_R) and L^2(Omega_R), respectively. Here v^R, p^R are the approximating solutions on the truncated domain Omega_R, the parameter R controls the exhausting of Omega. The artificial boundary conditions involve the Steklov-Poincare operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R^{-N}), where N can be arbitrarily large.

Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

Lagrangian approximations and weak solutions of the Navier-Stokes equations

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity containing a convergent subsequence with limit function v such that v is a weak solution of the Navier-Stokes equations.

Approximate approximations for the Poisson and the Stokes equations

The method of approximate approximations is based on generating functions representing an approximate partition of the unity, only. In the present paper this method is used for the numerical solution of the Poisson equation and the Stokes system in R^n (n = 2, 3). The corresponding approximate volume potentials will be computed explicitly in these cases, containing a one-dimensional integral, only. Numerical simulations show the efficiency of the method and confirm the expected convergence of essentially second order, depending on the smoothness of the data.

Functions satisfying holonomic q-differential equations

In a similar manner as in some previous papers, where explicit algorithms for finding the differential equations satisfied by holonomic functions were given, in this paper we deal with the space of the q-holonomic functions which are the solutions of linear q-differential equations with polynomial coefficients. The sum, product and the composition with power functions of q-holonomic functions are also q-holonomic and the resulting q-differential equations can be computed algorithmically.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

The Navier-Stokes Equations with Particle Methods

The non-stationary nonlinear Navier-Stokes equations describe the motion of a viscous incompressible fluid flow for 0equations are well-posed or not. Therefore we use a particle method to develop a system of approximate equations. We show that this system can be solved uniquely and globally in time and that its solution has a high degree of spatial regularity. Moreover we prove that the system of approximate solutions has an accumulation point satisfying the Navier-Stokes equations in a weak sense.