The diploma thesis included of informations of a Mathematical Olympiad, equations and systems of equations. The theoretical part contains a historical development and its organization, next it contains informations about equations and their teaching as it is given in the National Curriculum. The practical part consists of a collection of exercises on the topic of equations and systems of equations, which were included in the competition Mathematical Olympiad. Examples in the collection are divided into categories vhich appeard, and ranked by difficulty. Further practical part includes evaluation works heets completed by students of Pedagogicial faculty, University of South Bohemia. This thesis may serve as preparation pupils on Mathematical Olympiad or as a collection of examples for talented pupils

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In this thesis, we consider the computational methods for linear parabolic optimal control problems. We wish to minimize the cost functional while fulfilling the parabolic partial differential equations (PDE) constraint. This type of problems arises in many fields of science and engineering. Since solving such parabolic PDE optimal control problems often lead to a demanding computational cost and time, an effective algorithm is desired. In this research, we focus on the distributed control problems. Three types of cost functional are considered: Target States problems, Tracking problems, and All-time problems. Our major contribution in this research is that we developed a preconditioner for each kind of problems, so our iterative method is accelerated. In chapter 1, we gave a brief introduction to our problems with a literature review. In chapter 2, we demonstrated how to derive the first-order optimality conditions from the parabolic optimal control problems. Afterwards, we showed how to use the shooting method along with the flexible generalized minimal residual to find the solution. In chapter 3, we offered three preconditioners to enhance our shooting method for the problems with symmetric differential operator. Next, in chapter 4, we proposed another three preconditioners to speed up our scheme for the problems with non-symmetric differential operator. Lastly, we have the conclusion and the future development in chapter 5.

It has been shown that backward doubly stochastic differential equations (BDSDEs) provide a probabilistic representation for a certain class of nonlinear parabolic stochastic partial differential equations (SPDEs). It has also been shown that the solution of a BDSDE with Lipschitz coefficients can be approximated by first discretizing time and then calculating a sequence of conditional expectations. Given fixed points in time and space, this approximation has been shown to converge in mean square. In this thesis, we investigate the approximation of solutions of BDSDEs with coefficients that are measurable in time and space using a time discretization scheme with a view towards applications to SPDEs. To achieve this, we require the underlying forward diffusion to have smooth coefficients and we consider convergence in a norm which includes a weighted spatial integral. This combination of smoother forward coefficients and weaker norm allows the use of an equivalence of norms result which is key to our approach. We additionally take a brief look at the approximation of solutions of a class of infinite horizon BDSDEs with a view towards approximating stationary solutions of SPDEs. Whilst we remain agnostic with regards to the implementation of our discretization scheme, our scheme should be amenable to a Monte Carlo simulation based approach. If this is the case, we propose that in addition to being attractive from a performance perspective in higher dimensions, such an approach has a potential advantage when considering measurable coefficients. Specifically, since we only discretize time and effectively rely on simulations of the underlying forward diffusion to explore space, we are potentially less vulnerable to systematically overestimating or underestimating the effects of coefficients with spatial discontinuities than alternative approaches such as finite difference or finite element schemes that do discretize space. Another advantage of the BDSDE approach is that it is possible to derive an upper bound on the error of our method for a fairly broad class of conditions in a single analysis. Furthermore, our conditions seem more general in some respects than is typically considered in the SPDE literature.

The equation of motion for the flows of incompressible Newtonian fluids (Navier Stokes equations) under no-slip boundary conditions have been studied deeply from many perspectives. The questions of existence and uniqueness of both classical and weak solutions have received more than a fair share of attention. In this study the same problem for non-Newtonian fluids of second grade has been studied from the point of view of weak solutions and classical solutions for non-homogeneous boundary data, i.e., dynamical boundary conditions in regions with permeable boundaries. We consider the situation where a container is immersed in a larger fluid body and the boundary admits fluid particles moving across it in the direction of the normal. In this study we give alternative approaches through formulations of' dynamics at the boundary', the idea being that the normal component of velocity at the boundary is viewed as an unknown function which satisfies a differential equation intricately coupled to the flow in the region 'enclosed' by the boundary. We describe two mathematical models denoted by Problem PI and Problem P2. These models lead to dynamics at a permeable boundary, and a kinematical boundary condition for normal flow through the boundary. These conditions take into account the curvature of the boundary which enforces certain stresses. We then show with the help of the energy method that for fluids of second grade, the dynamics at the boundary and the boundary condition lead to conditional stability of the rest state for Problem P1 and Problem P2. We also prove uniqueness of classical solutions for the two models. The existence of a weak solution for this system of evolution equations is proved only for Problem P2 with the help of the Faedo-Galerkin method with a special basis. In this case the special basis is formed by eigenfunctions. The existence proof of at least one classical solution, local in time is established by means of a version of the Fixed-point Theorem of Bohnenblust and Karlin, and the Ascoli-Arzela Theorem. / Thesis (PhD (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

Stochastic delay differential equations naturally arise in models of complex natural phenomena, yet continue to resist efforts to find analytical solutions to them: general solutions are limited to linear systems with additive noise and a single delayed term. In this work we solve the case of distributed delays in linear systems with additive noise. Key to our solution is the development of a consistent interpretation for integrals over stochastic variables, obtained by means of a virtual discretization procedure. This procedure makes no assumption on the form of noise, and would likely be useful for a wider variety of cases than those we have considered. We show how it can be used to map the distributed delay equation to a known multivariate system, and obtain expressions for the system's time-dependent mean and autocovariance. These are in the form of series over the system's natural modes and completely define the solution. — An interpretation of the system as an amplitude process is explored. We show that for a wide range of realistic parameters, dynamics are dominated by only a few modes, implying that most of the observed behaviour of stochastic delayed equations is constrained to a low-dimensional subspace. — The expression for the autocovariance is given particular attention. A recurring problem for stochastic delay equations is the description of their temporal structure. We show that the series expression for the autocovariance does converge over a meaningful range of time lags, and therefore provides a means of describing this temporal structure.

Let H be a separable Hilbert space. Suppose (Ω, F, Ft, P) is a complete stochastic basis with a right continuous filtration and {Wt,t ∈ R} is an H-valued cylindrical Brownian motion with respect to {Ω, F, Ft, P). U(t, s) denotes an almost strong evolution operator generated by a family of unbounded closed linear operators on H. Consider the semilinear stochastic integral equation [formula omitted] where • f is of monotone type, i.e., ft(.) = f(t, w,.) : H → H is semimonotone, demicon-tinuous, uniformly bounded, and for each x ∈ H, ft(x) is a stochastic process which satisfies certain measurability conditions. • gs(.) is a uniformly-Lipschitz predictable functional with values in the space of Hilbert-Schmidt operators on H. • Vt is a cadlag adapted process with values in H. • X₀ is a random variable. We obtain existence, uniqueness, boundedness of the solution of this equation. We show the solution of this equation changes continuously when one or all of X₀, f, g, and V are varied. We apply this result to find stationary solutions of certain equations, and to study the associated large deviation principles. Let {Zt,t ∈ R} be an H-valued semimartingale. We prove an Ito-type inequality and a Burkholder-type inequality for stochastic convolution [formula omitted]. These are the main tools for our study of the above stochastic integral equation. / Science, Faculty of / Mathematics, Department of / Graduate

Collocation with cubic splines is used as a method for solving Linear second order parabolic partial differential equations. The collocation method is shown to be equivalent to a finite difference method that is consistent with the differential equation and stable in the sense of Von Neumann. Results of numerical computations are given, as well as an application of the method to a moving boundary problem for the heat equation. / Science, Faculty of / Mathematics, Department of / Graduate

The asymptotic behaviour of nonoscillatory solutions of second order nonlinear ordinary differential equations is studied. Necessary and sufficient conditions are given for the existence of positive solutions with specified asymptotic behaviour at infinity. Existence of nonoscillatory solutions is established using the Schauder-Tychonoff fixed point theorem. Techniques such as factorization of linear disconjugate operators are employed to reveal the similar nature of asymptotic solutions of nonlinear differential equations to that of linear equations. Some examples illustrating the asymptotic theory of ordinary differential equations are given. / Science, Faculty of / Mathematics, Department of / Graduate

In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.