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.

A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Solution properties of the de Branges differential recurrence equation

In this 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums thas was published by Askey and Gasper in 1976. The de Branges functions Tn/k(t) are defined as the solutions of a system of differential recurrence equations with suitably given initial values. The essential fact used in the proof of the Bieberbach and Milin conjectures is the statement Tn/k(t)<=0. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λn/k(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', Tn/k(t)=-kΛn/k(t), and the positivity results in both proofs Tn/k(t)<=0 are essentially the same. In this paper we study differential recurrence equations equivalent to de Branges' original ones and show that many solutions of these differential recurrence equations don't change sign so that the above inequality is not as surprising as expected. Furthermore, we present a multiparameterized hypergeometric family of solutions of the de Branges differential recurrence equations showing that solutions are not rare at all.

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.

Total differential scattering cross section of {Ar^+}-Ar at 15 to 400 keV

We report on the measurement of the total differential scattering cross section of {Ar^+}-Ar at laboratory energies between 15 and 400 keV. Using an ab initio relativistic molecular program which calculates the interatomic potential energy curve with high accuracy, we are able to reproduce the detailed structure found in the experiment.

An introduction to ordinary differential equations by computer algebra-systems

We report on an elementary course in ordinary differential equations (odes) for students in engineering sciences. The course is also intended to become a self-study package for odes and is is based on several interactive computer lessons using REDUCE and MATHEMATICA . The aim of the course is not to do Computer Algebra (CA) by example or to use it for doing classroom examples. The aim ist to teach and to learn mathematics by using CA-systems.

Root parametrized differential equations

The present thesis is about the inverse problem in differential Galois Theory. Given a differential field, the inverse  problem asks which linear algebraic groups can be realized as differential Galois groups of Picard-Vessiot extensions of this field.   In this thesis we will concentrate on the realization of the classical groups as differential Galois groups. We introduce a method for a very general realization of these groups. This means that we present for the classical groups of Lie rank $l$ explicit linear differential equations where the coefficients are differential polynomials in $l$ differential indeterminates over an algebraically closed field of constants $C$, i.e. our differential ground field is purely differential transcendental over the constants.   For the groups of type $A_l$, $B_l$, $C_l$, $D_l$ and $G_2$ we managed to do these realizations at the same time in terms of Abhyankar's program 'Nice Equations for Nice Groups'. Here the choice of the defining matrix is important. We found out that an educated choice of $l$ negative roots for the parametrization together with the positive simple roots leads to a nice differential equation and at the same time defines a sufficiently general element of the Lie algebra. Unfortunately for the groups of type $F_4$ and $E_6$ the linear differential equations for such elements are of enormous length. Therefore we keep in the case of $F_4$ and $E_6$ the defining matrix differential equation which has also an easy and nice shape.   The basic idea for the realization is the application of an upper and lower bound criterion for the differential Galois group to our parameter equations and to show that both bounds coincide. An upper and lower bound criterion can be found in literature. Here we will only use the upper bound, since for the application of the lower bound criterion an important condition has to be satisfied. If the differential ground field is $C_1$, e.g., $C(z)$ with standard derivation, this condition is automatically satisfied. Since our differential ground field is purely differential transcendental over $C$, we have no information whether this condition holds or not.   The main part of this thesis is the development of an alternative lower bound criterion and its application. We introduce the specialization bound. It states that the differential Galois group of a specialization of the parameter equation is contained in the differential Galois group of the parameter equation. Thus for its application we need a differential equation over $C(z)$ with given differential Galois group. A modification of a result from Mitschi and Singer yields such an equation over $C(z)$ up to differential conjugation, i.e. up to transformation to the required shape. The transformation of their equation to a specialization of our parameter equation is done for each of the above groups in the respective transformation lemma.

Generic Finite Element Model for Mechanically Consistent Scaling of Composite Beam-to-column Joint with Welded Connection

To study the behaviour of beam-to-column composite connection more sophisticated finite element models is required, since component model has some severe limitations. In this research a generic finite element model for composite beam-to-column joint with welded connections is developed using current state of the art local modelling. Applying mechanically consistent scaling method, it can provide the constitutive relationship for a plane rectangular macro element with beam-type boundaries. Then, this defined macro element, which preserves local behaviour and allows for the transfer of five independent states between local and global models, can be implemented in high-accuracy frame analysis with the possibility of limit state checks. In order that macro element for scaling method can be used in practical manner, a generic geometry program as a new idea proposed in this study is also developed for this finite element model. With generic programming a set of global geometric variables can be input to generate a specific instance of the connection without much effort. The proposed finite element model generated by this generic programming is validated against testing results from University of Kaiserslautern. Finally, two illustrative examples for applying this macro element approach are presented. In the first example how to obtain the constitutive relationships of macro element is demonstrated. With certain assumptions for typical composite frame the constitutive relationships can be represented by bilinear laws for the macro bending and shear states that are then coupled by a two-dimensional surface law with yield and failure surfaces. In second example a scaling concept that combines sophisticated local models with a frame analysis using a macro element approach is presented as a practical application of this numerical model.

On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials

In this work, we have mainly achieved the following: 1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved; 2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable; 3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials; 4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices. Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue.