This thesis deals with hypergeometric functions in several complex variables and systems of partial differential equations of hypergeometric type. One of the main objects of study in the thesis is the so-called Horn system of equations: xiPi(θ)y(x) = Qi(θ)y(x), i = 1, ..., n.Here x ∈ℂn, θ = (θ1, ..., θn),θi = xi ∂/∂xi , Pi and Qi are nonzero polynomials. By definition hypergeometric functions are (multi-valued) analytic solutions to this system of equations. The main purpose of the thesis is to systematically investigate the Horn system of equations and properties of its solutions.To construct solutions to the Horn system we use one of the variants of the Laplace transform which leads to a system of linear difference equations with polynomial coefficients. Solving this system we represent a solution to the Horn system in the form of an iterated Puiseux series.We give an explicit formula for the dimension of the space of analytic solutions to the Horn system at a generic point under some assumptions on its parameters. The proof is based on the study of the module over the Weyl algebra of linear differential operators with polynomial coefficients associated with the Horn system. Combining this formula with the theorem which allows one to represent a solution to the Horn system in the form of an iterated Puiseux series, we obtain a basis in the space of analytic solutions to this system of equations.Another object of study in the thesis is the singular set of a nonconuent hypergeometric function in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of the Newton polytopes of their defining polynomials. In particular we obtain a geometric description of the zero set of the discriminant of a general algebraic equation.In the case of two variables one can say much more about singularities of nonconuent hypergeometric functions. We give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconuent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials whose zero loci contain the singularities of the factors.

This thesis consists of four papers which are all related to the regularity properties of free boundary problems. The problems considered have in common that they have some sort of two-phase behaviour.In papers I-III we study the interior regularity of different two-phase free boundary problems. Paper I is mainly concerned with the regularity properties of the free boundary, while in papers II and III we devote our study to the regularity of the function, but as a by-product we obtain some partial regularity of the free boundary.The problem considered in paper IV has a somewhat different nature. Here we are interested in certain approximations of the obstacle problem. Two major differences are that we study regularity properties close to the fixed boundary and that the problem converges to a one-phase free boundary problem. / QC 20100728

This thesis concerns the analysis and numerical simulation of wave propagation problems described by systems of linear hyperbolic partial differential equations. A major challenge in wave propagation problems is numerical simulation of high frequency waves. When the wavelength is very small compared to the overall size of the computational domain, we encounter a multiscale problem. Examples include the forward and the inverse seismic wave propagation, radiation and scattering problems in computational electromagnetics and underwater acoustics. In direct numerical simulations, the accuracy of the approximate solution is determined by the number of grid points or elements per wavelength. The computational cost to maintain constant accuracy grows algebraically with the frequency, and for sufficiently high frequency, direct numerical simulations are no longer feasible. Other numerical methods are therefore needed. Asymptotic methods, for instance, are good approximations for very high frequency waves. They are based on constructing asymptotic expansions of the solution. The accuracy increases with increasing frequency for a fixed computational cost. Most asymptotic techniques rely on geometrical optics equations with frequency independent unknowns. There are however two deficiencies in the geometrical optics solution. First, it does not include diffraction effects. Secondly, it breaks down at caustics. Geometrical theory of diffraction provides a technique for adding diffraction effects to the geometrical optics approximation by introducing diffracted rays. In papers 1 and 2 we present a numerical algorithm for computing an important type of diffracted rays known as creeping rays. Another asymptotic model which is valid also at caustics is based on Gaussian beams. In papers 3 and 4, we present an error analysis of Gaussian beams approximation and develop a new numerical algorithm for computing Gaussian beams, respectively. Another challenge in computation of wave propagation problems arises when the system of equations consists of second order hyperbolic equations involving mixed space-time derivatives. Examples include the harmonic formulation of Einstein’s equations and wave equations governing elasticity and acoustics. The classic computational treatment of such second order hyperbolic systems has been based on reducing the systems to first order differential forms. This treatment has however the disadvantage of introducing auxiliary variables with their associated constraints and boundary conditions. In paper 5, we treat the problem in the second order differential form, which has advantages for both computational efficiency and accuracy over the first order formulation. Finally, paper 6 concerns the concept of well-posedness for a class of linear hyperbolic initial boundary value problems which are not boundary stable. The well-posedness is well established for boundary stable hyperbolic systems for which we can obtain sharp estimates of the solution including estimates at boundaries. There are, however, problems which are not boundary stable but are well-posed in a weaker sense, i.e., the problems for which an energy estimate can be obtained in the interior of the domain but not on the boundaries. We analyze a model problem of this type. Possible applications arise in elastic wave equations and Maxwell’s equations describing glancing and surface waves. / QC 20100830

his work is concentrated on efforts to efficiently compute properties of systems, modelled by differential equations, involving multiple scales. Goal oriented adaptivity is the common approach to all the treated problems. Here the goal of a numerical computation is to approximate a functional of the solution to the differential equation and the numerical method is adapted to this task. The thesis consists of four papers. The first three papers concern the convergence of adaptive algorithms for numerical solution of differential equations; based on a posteriori expansions of global errors in the sought functional, the discretisations used in a numerical solution of the differential equiation are adaptively refined. The fourth paper uses expansion of the adaptive modelling error to compute a stochastic differential equation for a phase-field by coarse-graining molecular dynamics. An adaptive algorithm aims to minimise the number of degrees of freedom to make the error in the functional less than a given tolerance. The number of degrees of freedom provides the convergence rate of the adaptive algorithm as the tolerance tends to zero. Provided that the computational work is proportional to the degrees of freedom this gives an estimate of the efficiency of the algorithm. The first paper treats approximation of functionals of solutions to second order elliptic partial differential equations in bounded domains of ℝd, using isoparametric $d$-linear quadrilateral finite elements. For an adaptive algorithm, an error expansion with computable leading order term is derived %. and used in a computable error density, which is proved to converge uniformly as the mesh size tends to zero. For each element an error indicator is defined by the computed error density multiplying the local mesh size to the power of 2+d. The adaptive algorithm is based on successive subdivisions of elements, where it uses the error indicators. It is proved, using the uniform convergence of the error density, that the algorithm either reduces the maximal error indicator with a factor or stops; if it stops, then the error is asymptotically bounded by the tolerance using the optimal number of elements for an adaptive isotropic mesh, up to a problem independent factor. Here the optimal number of elements is proportional to the d/2 power of the Ldd+2 quasi-norm of the error density, whereas a uniform mesh requires a number of elements proportional to the d/2 power of the larger L1 norm of the same error density to obtain the same accuracy. For problems with multiple scales, in particular, these convergence rates may differ much, even though the convergence order may be the same. The second paper presents an adaptive algorithm for Monte Carlo Euler approximation of the expected value E[g(X(τ),\τ)] of a given function g depending on the solution X of an \Ito\ stochastic differential equation and on the first exit time τ from a given domain. An error expansion with computable leading order term for the approximation of E[g(X(T))] with a fixed final time T>0 was given in~[Szepessy, Tempone, and Zouraris, Comm. Pure and Appl. Math., 54, 1169-1214, 2001]. This error expansion is now extended to the case with stopped diffusion. In the extension conditional probabilities are used to estimate the first exit time error, and difference quotients are used to approximate the initial data of the dual solutions. For the stopped diffusion problem the time discretisation error is of order N-1/2 for a method with N uniform time steps. Numerical results show that the adaptive algorithm improves the time discretisation error to the order N-1, with N adaptive time steps. The third paper gives an overview of the application of the adaptive algorithm in the first two papers to ordinary, stochastic, and partial differential equation. The fourth paper investigates the possibility of computing some of the model functions in an Allen--Cahn type phase-field equation from a microscale model, where the material is described by stochastic, Smoluchowski, molecular dynamics. A local average of contributions to the potential energy in the micro model is used to determine the local phase, and a stochastic phase-field model is computed by coarse-graining the molecular dynamics. Molecular dynamics simulations on a two phase system at the melting point are used to compute a double-well reaction term in the Allen--Cahn equation and a diffusion matrix describing the noise in the coarse-grained phase-field. / QC 20100823

In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented. After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diusion equations.

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