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In this thesis we focus on the development and investigation of methods for the computation of confluent hypergeometric functions. We point out the relations between these functions and parabolic boundary value problems and demonstrate applications to models of heat transfer and fluid dynamics. For the computation of confluent hypergeometric functions on compact (real or complex) intervals we consider a series expansion based on the Hadamard product of power series. It turnes out that the partial sums of this expansion are easily computable and provide a better rate of convergence in comparison to the partial sums of the Taylor series. Regarding the computational accuracy the problem of cancellation errors is reduced considerably. Another important tool for the computation of confluent hypergeometric functions are recurrence formulae. Although easy to implement, such recurrence relations are numerically unstable e.g. due to rounding errors. In order to circumvent these problems a method for computing recurrence relations in backward direction is applied. Furthermore, asymptotic expansions for large arguments in modulus are considered. From the numerical point of view the determination of the number of terms used for the approximation is a crucial point. As an application we consider initial-boundary value problems with partial differential equations of parabolic type, where we use the method of eigenfunction expansion in order to determine an explicit form of the solution. In this case the arising eigenfunctions depend directly on the geometry of the considered domain. For certain domains with some special geometry the eigenfunctions are of confluent hypergeometric type. Both a conductive heat transfer model and an application in fluid dynamics is considered. Finally, the application of several heat transfer models to certain sterilization processes in food industry is discussed.

In this thesis, we investigate the quantization problem of Gaussian measures on Banach spaces by means of constructive methods. That is, for a random variable X and a natural number N, we are searching for those N elements in the underlying Banach space which give the best approximation to X in the average sense. We particularly focus on centered Gaussians on the space of continuous functions on [0,1] equipped with the supremum-norm, since in that case all known methods failed to achieve the optimal quantization rate for important Gauss-processes. In fact, by means of Spline-approximations and a scheme based on the Best-Approximations in the sense of the Kolmogorov n-width we were able to attain the optimal rate of convergence to zero for these quantization problems. Moreover, we established a new upper bound for the quantization error, which is based on a very simple criterion, the modulus of smoothness of the covariance function. Finally, we explicitly constructed those quantizers numerically.

Considering the numerical simulation of mathematical models it is necessary to have efficient methods for computing special functions. We will focus our considerations in particular on the classes of Mittag-Leffler and confluent hypergeometric functions. The PhD Thesis can be structured in three parts. In the first part, entire functions are considered. If we look at the partial sums of the Taylor series with respect to the origin we find that they typically only provide a reasonable approximation of the function in a small neighborhood of the origin. The main disadvantages of these partial sums are the cancellation errors which occur when computing in fixed precision arithmetic outside this neighborhood. Therefore, our aim is to quantify and then to reduce this cancellation effect. In the next part we consider the Mittag-Leffler and the confluent hypergeometric functions in detail. Using the method we developed in the first part, we can reduce the cancellation problems by "modifying" the functions for several parts of the complex plane. Finally, in in the last part two other approaches to compute Mittag-Leffler type and confluent hypergeometric functions are discussed. If we want to evaluate such functions on unbounded intervals or sectors in the complex plane, we have to consider methods like asymptotic expansions or continued fractions for large arguments z in modulus.

The present work considers the normal approximation of the binomial distribution and yields estimations of the supremum distance of the distribution functions of the binomial- and the corresponding standardized normal distribution. The type of the estimations correspond to the classical Berry-Esseen theorem, in the special case that all random variables are identically Bernoulli distributed. In this case we state the optimal constant for the Berry-Esseen theorem. In the proof of these estimations several inequalities regarding the density as well as the distribution function of the binomial distribution are presented. Furthermore in the estimations mentioned above the distribution function is replaced by the probability of arbitrary, not only unlimited intervals and in this new situation we also present an upper bound.