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.

A Conservative Front Tracking Algorithm

The discontinuities in the solutions of systems of conservation laws are widely considered as one of the difficulties in numerical simulation. A numerical method is proposed for solving these partial differential equations with discontinuities in the solution. The method is able to track these sharp discontinuities or interfaces while still fully maintain the conservation property. The motion of the front is obtained by solving a Riemann problem based on the state values at its both sides which are reconstructed by using weighted essentially non oscillatory (WENO) scheme. The propagation of the front is coupled with the evaluation of "dynamic" numerical fluxes. Some numerical tests in 1D and preliminary results in 2D are presented.

Exercise sheet 4

Exercises and solutions in LaTex

MA311exam 1998

Exam questions and solutions in LaTex

Exercise sheet 4 pdf

Exercises and solutions in PDF

Exercise sheet 3 pdf

Exercises and solutions in PDF

Exercise sheet 3

Exercises and solutions in LaTex

MA311exam 1998 pdf

Exam questions and solutions in PDF

Differential Equations II

Exercises and solutions for a second year differential equations course.

Applications of Partial Differentiation

Exercises and solutions for an applications of partial differentiation course. Diagrams for the questions are all together in the support.zip file, as .eps files

Ecuaciones diferenciales estocásticas con condición final y soluciones de viscosidad de EDPS semilineales de segundo orden

El objetivo de este documento es recopilar algunos resultados clasicos sobre existencia y unicidad ´ de soluciones de ecuaciones diferenciales estocasticas (EDEs) con condici ´ on final (en ingl ´ es´ Backward stochastic differential equations) con particular enfasis en el caso de coeficientes mon ´ otonos, y su cone- ´ xion con soluciones de viscosidad de sistemas de ecuaciones diferenciales parciales (EDPs) parab ´ olicas ´ y el´ıpticas semilineales de segundo orden.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

Local roles of TGF-beta superfamily members in the control of ovarian follicle development

Members of the transforming growth factor-beta (TGF-beta) superfamily have wide-ranging influences on many tissue and organ systems including the ovary. Two recently discovered TGF-beta superfamily members, growth/differentiation factor-9 (GDF-9) and bone morphogenetic protein-15 (BMP-15; also designated as GDF-9B) are expressed in an oocyte-specific manner from a very early stage and play a key role in promoting follicle growth beyond the primary stage. Follicle growth to the small antral stage does not require gonadotrophins but appears to be driven by local autocrine/paracrine signals from both somatic cell types (granulosa and theca) and from the oocyte. TGF-beta superfamily members expressed by follicular cells and implicated in this phase of follicle development include TGF-beta, activin, GDF-9/9B and several BMPs. Acquisition of follicle-stimulating hormone (FSH) responsiveness is a pre-requisite for growth beyond the small antral stage and evidence indicates an autocrine role for granulosa-derived activin in promoting granulosa cell proliferation, FSH receptor expression and aromatase activity. Indeed, some of the effects of FSH on granulosa cells may be mediated by endogenous activin. At the same time, activin may act on theca cells to attenuate luteinizing hormone (LH)-dependent androgen production in small to medium-size antral follicles. Dominant follicle selection appears to depend on differential FSH sensitivity amongst a growing cohort of small antral follicles. Activin may contribute to this selection process by sensitizing those follicles with the highest "activin tone" to FSH. Production of inhibin, like oestradiol, increases in selected dominant follicles, in an FSH- and insulin-like growth factor-dependent manner and may exert a paracrine action on theca cells to upregulate LH-induced secretion of androgen, an essential requirement for further oestradiol secretion by the pre-ovulatory follicle. Like activin, BMP-4 and -7 (mostly from theca), and BMP-6 (mostly from oocyte), can enhance oestradiol and inhibin secretion by bovine granulosa cells while suppressing progesterone secretion; this suggests a functional role in delaying follicle luteinization and/or atresia. Follistatin, on the other hand, may favor luteinization and/or atresia by bio-neutralizing intrafollicular activin and BMPs. Activin receptors are expressed by the oocyte and activin may have a further intrafollicular role in the terminal stages of follicle differentiation to promote oocyte maturation and developmental competence. In a reciprocal manner, oocyte-derived GDF-9/9B may act on the surrounding cumulus granulosa cells to attenuate oestradiol output and promote progesterone and hyaluronic acid production, mucification and cumulus expansion.(C) 2003 Elsevier Science B.V. All rights reserved.

On the system of rational difference equations x(n) = A+(1)over-bar y(n-p), y(n) = A+(gamma(n-1))over-bar x(n-r)y(n-5)

In this paper, we study the behavior of the positive solutions of the system of two difference equations [GRAPHICS] where p >= 1, r >= 1, s >= 1, A >= 0, and x(1-r), x(2-r),..., x(0), y(1-max) {p.s},..., y(0) are positive real numbers. (c) 2005 Elsevier Inc. All rights reserved.