Capstone Projects
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Item Open Access A Short Note On Solving 1-D Porous Medium Equation by Finite Element Methods(Nazarbayev University School of Science and Technology, 2016-05-20) Matayev, ChingisPorous Medium Equation (PME) is one of the simplest types of of nonlinear evolution equation of parabolic type. It emerges in the description of di erent natural phenomena, and its theory and properties depart strongly from the heat equation, ut = u, its most famous relative. Hence the interest of its study, both for the pure mathematician and the applied scientist (Vazquez, 2006). The aim of this paper is to study the Porous Medium Equation in one dimension.Item Open Access Actuarial Applications of a two-parameter generalized Logistic model(Nazarbayev University School of Science and Technology, 2018-05-10) Alekberova, NargizThis project presents a generalization of the classical logistic and the Gompertz model by using two parameter power law exponent. The suggested generalized with two parameters models was shown by numerical example to be potentially a better choice for fitting a certain data. The solutions of these models are used in terms of force of mortality function in actuarial math as insurance contingency models or as hazard function for quality control in science and engineering. The complexity of solution is presented in the form of hypergeometric function and corresponding to it nonlinear implicit regression analysis. Due to the mentioned complexity we develop the approximation of the actual solution and consider only its special cases.Item Open Access Analysis of Dynamic Pull-in for a Graphene-based MEMS Model(Nazarbayev University School of Science and Technology, 2018) Omarov, DaniyarA novel procedure based on the Sturm’s theorem for real-valued polynomials is developed to predict and identify periodic solutions and non-periodic solutions in the pull-in analysis of a graphene-based MEMS lumped parameter model with general initial conditions. It is demonstrated that under specific conditions on the lumped parameters and the initial conditions, the model has certain periodic solutions and otherwise there is no such solutions. This theoretical procedure is made practical by numerical implementations with Python scripts to verify the predicted behaviour of the periodic solutions. Numerical simulations are performed with sample data to justify by this procedure the analytically predicted existence of periodic solutions. Also, Low Order Fourier Approximation is used to find the solution for the linear spring case. Comparison with the highly accurate Runge-Kutta method is done to verify derived values from the new numerical approximation.Item Open Access Analytic Solutions for a Nonlinear Transport Equation(Nazarbayev University School of Science and Technology, 2019-08-07) Biyar, Magzhan; da Silva, Daniel Oliveira; Castro Castilla, Alejandro JavierWe prove that the Cauchy problem for a transport equation with algebraic nonlinearity of degree p with initial data in Gevrey spaces is locally well-posed. In particular, we show that the analyticity of solutions persists for a short time and we derive a sufficient condition for solutions to be analytic for all times.Item Open Access Boundary Element Method for Stokes Flow in Incompressible Newtonian Fluids(Nazarbayev University School of Science and Technology, 2015-04) Tazhimbetov, Nurbek AkhmetulyIn this thesis, I present different discretization techniques for boundary integral method for Stokes flow in case of an incompressible Newtonian fluid. Boundary integral method (BIM) is one of many techniques that are used to solve Partial Differencial Equations (PDE) numerically. However, the basic advantage of the BIM is that it reduces the problem from n-dimensional domain to n - 1; for example, the two-dimensional square-box that contains viscous liquid can be solved by using the values of an unkown function at the boundary of square. Nevertheless, the BIM exhibits some challenges in finding the Green's function for a particular domain or differential operator, solving the integral equations and, especially, in computing the values of a complex domain. The latter one is quite diffcult because the flow diverges at corners (exhibits singularity). The goal of this work is to derive general analytical solution for Stokes equation (in integral equations form) and to compute the discretized integral equations using different quadrature rules for cavity problem.Item Open Access Copula functions in Credit Metrics’ VaR estimation(Nazarbayev University School of Science and Technology, 2019-08-08) Magzanov, Shynggys; Wei, Dongming; Assylbekov, ZhenisbekCredit risk modelling of a portfolio of exposures is essential part of activity of every financial institution. However this procedure is complicated since the joint behavior of chosen exposures must be known. In this paper Value at Risk percentile of the portfolio consisting of three corporate bonds issued by Lukoil, Gazprom and Norilsk Nickel was estimated at three different significance levels within the frame of Credit Metrics approach proposed by J.P.Morgan. Following the Asset value model, Monte-Carlo simulations were performed to obtain possible portfolio values in one year time horizon. Where the joint distribution of asset returns of three companies was constructed by means of pair-copula construction method discussed in Aas, Czado, Frigessi,Bakken (2009). Results reveal that for particular portfolio of bonds at 90%, 95% and 99% confidence levels the value of our portfolio will not fall below 2057.915 ,1798.117 and 1375.011 dollars respectively.Item Open Access Copulas and Correlation in Statistical Risk Theory(Nazarbayev University School of Science and Technology, 2018) Rat, SaraThe Financial Risk Management (FRM) aims to identify, measure and manage risks in different sectors. One of the core things during such operations is measuring different dependencies. Linear correlation is known as one of the most popular measure of dependence, however it is known that it is a reasonable mea- sure of dependence only when variables are Normally distributed, but this is not the case with credit and portfolio risks, therefore other measures of dependency are needed. This paper presents a Copula function for bivariate case as one of the tools to analyze dependencies in portfolio risk management. Copulas were first introduced by Sklar in 1959 [8], and in 1999 they were studied in financial context for the first time by Embrechts et al. in 1999 [4]. Motivated by the copula analysis of European stock portfolios [6], this paper aims to analyze portfolio consisting of Asian S&P Asia 50 and S&P BSE 100 indices, and apply copula to this portfolio.Item Open Access Estimation and application of best ARIMA model for forecasting the uranium price.(Nazarbayev University School of Science and Technology, 2018-05-13) Amangeldi, MedeuThis paper presents the application of an iterative approach for prediction of uranium price by model identification, parameter estimation and diagnostic checking which are designed by Box and Jenkins. In particular, the autoregressive integrated moving average model is used to predict the future values of monthly uranium price. As the analysis of structural dependence in observations is one of the key features of time series analysis, the past values, which were taken as monthly values from January 2000 to June 2017, are used for forecasting. As a result, ARIMA (2,1,0) became one that met all the criteria and predicted the increase of uranium price over time within 95% confidence.Item Open Access Finite Element Solution of Thermal Gee-Lyon Flows in a Circular Tube(Nazarbayev University School of Science and Technology, 2016-05) Aimambet, NarkenThe goal of this project is to calculate the temperature distribu- tion of certain pressure-driven non-Newtonian ows inside a circular tube. The rheology under consideration is the type in which the shear stress is an implicit function of the shear rate de ned as the inverse function of an odd function. The uid velocity in the tube is approximated by the steady state velocity pro le along the tube length and the viscosity is assumed to be independent of the temperature. The velocity pro le is computed by using Mathematica's build-in ODE solver semi-analytically. The corresponding steady state temperature pro le at tube length is then calculated taking into account of heat source generated by shear rate from the uid ow by solving an ODE. The temperature distribution from the entrance to the fully developed region is then approximated numerically by using the axisymmet- ric linear triangular nite elements. Material and geometric constants and data for extrusion of chemical Lucite through the tube in literature are used for the numerical example. Comparison of the numerical result with the industrial experimental result is made at a point along the central axis of the tube.Item Open Access Finite element solutions of the nonlinear RAPM Black-Scholes model(Nazarbayev University School of Science and Technology, 2016-05-09) Zhexembay, Laila; Pak, AndreyThe main purpose of our Capstone project is to study the Risk-Adjusted Pricing Methodology (RAMP) Black-Scholes model and to find the finite element solutions of the nonlinear Black-Scholes equation. The RAPM is one of the many nonlinear models in option pricing considering factors which affect the volatility in the original Black-Scholes equation. This model can be simplifed to a nonlinear parabolic equation in a new vari- able which equals the product of Gamma and the price of the underlying asset. Galerkin nite element method is applied to the parabolic equation. Two types of solutions will be presented: one using the linear elements and the other using quadratic elements. Local finite element equations for the linear and quadratic elements are derived with some specifc inter- polations of the nonlinear terms. Numerical solutions are obtained and compared to the results in literature. The explanation of the discrepancies will be given together with the future goals of this study.Item Open Access Frobenius Singularities of Algebraic Sets of Matrices(Nazarbayev University School of Science and Technology, 2019-08-07) Yerlanov, Madi; Kadyrsizova, Zhibek; Sica, FrancescoWhen one studies certain rings, it is natural to classify them according to certain properties. This project focuses on the study of properties of commutative rings associated with algebraic sets. In particular, we consider the algebraic set of pairs of square matrices whose commutator has a zero diagonal. We prove that it is irreducible and F-regular for matrices of all sizes and when the matrix entries are from a eld of positive prime characteristic. In addition, we provide a proof of its F-purity and nd a system of parameters on it. Moreover, we state several conjectures associated to this topic.Item Open Access Initial Explorations on Regularizing the SCRN Model(Nazarbayev University School of Science and Technology, 2018-05) Kabdolov, OlzhasRecurrent neural networks are very powerful sequence models which are used for language modeling as well. Under correct regularization such as naive dropout these models are able to achieve substantial improvement in their performance. We regularize the Structurally Constrained Recurrent Network (SCRN) model and show that despite its simplicity it can achieve the performance comparable to the ubiquitous LSTM model in language modeling task while being smaller in size and up to 2x faster to train. Further analysis shows that regularizing both context and hidden states of the SCRN is crucial.Item Open Access Modeling of corneal deformation under air-puff by nonlinear differential equations(Nazarbayev University School of Science and Technology, 2016-05) Jangabylova, Aliya; Zhalgas, AidanaIntraocular Pressure (IOP) is a main factor for the diagnosis of glaucoma. In this report, the kinematic viscoelastic corneal models, specifcally the Maxwell and the Kelvin-Voight models, of human eye ball will be proposed for determining the displacement of the cornea during the air-puff tonometry simulations and its relationship to IOP. The purpose of project is to study the in uence of elasticity and viscosity to the corneal deformations under an air puff.Item Open Access Multiple Point Compression on Elliptic Curves(Nazarbayev University School of Science and Technology, 2015) Otemissov, AdiletThe paper aims at developing new point compression algorithms which are useful in mobile communication systems where Elliptic Curve Cryptography is employed to achieve secure data storage and transmission. Compression algorithms allow elliptic curve points to be represented in the form that balances the usage of memory and computational power. The two- and three-point compression algorithms developed by Khabbazian, Gulliver and Bhargava [4] are reviewed and extended to generic cases of four and five points. The proposed methods use only basic operations (multiplication, division, etc.) and avoids square root finding. In addition, a new two-point compression method which is heavy in compression phase and light in decompression is developed.Item Open Access Nonlinear Regression Analysis of the generalized Logistic Model as an Actuarial life contingency model(Nazarbayev University School of Science and Technology, 2019-08-08) Kadenova, Aida; Wei, Dongming; Erlangga, YogiThe aim of this project is to analyze three different population models such as Gompertz, Logistic and Generalized Logistic based USA population data. Finding the appropriate model is essential in actuarial application. Firstly, the parameters of the two models are estimated using the special function ~nls in the R language program. But, due to some complexities, parameters of the generalized logistic model are evaluated using the new method from Causton`s paper. Secondly, two different comparison methods such as residual plot and AIC are used to analyze what model is appropriate for USA statistical data. Lastly, suitable models are used to estimate the force of mortality.Item Open Access Nonlinear Schrodinger Equation(Nazarbayev University School of Science and Technology, 2019-08-08) Kazbek, MoldirRogue waves are fascinating destructive phenomena in nature that have not been fully explained so far [1-3]. Oceanographers commonly agree that linear theories cannot provide explanations for their existence[6,7]. Only nonlinear theories can explain the dramatic concentration of energy into a single "wall of water" well above the average height of the surrounding waves[3,8,9]. Among nonlinear theories the most fundamental is based on the nonlinear Schr odinger equation (NLSE)[6]. If the fundamental approach allows us to give a basic explanation, then it can be extended to more general ones which take into account the two-dimensional nature of the problem. Which is our main goal.Item Open Access On a stochastic interacting particle system with pushing dynamics(Nazarbayev University School of Science and Technology, 2016) Abdukadyrov, NurlanIn this paper we study a stochastic two-particle system on Z where particles interact each other by pushing dynamics. We derive the explicit formulas of the transition probability and of the probability distributions of each particle's position at time t. Finally, we discuss about the generalization of our works to N-particle system.Item Open Access On logistic-normal distribution(Nazarbayev University School of Science and Technology, 2016-04) Almagambetova, Ayanna; Zakiyeva, NazgulExisting distributions do not always provide an adequate fit to the complex real world data. Hence, the interest in developing more flexible statistical distributions remains strong in statistics profession. In this project, we present a family of generalized normal distributions, the T-normal family. We study in some details a member of the proposed family namely, the logistic-normal (LN) distribution. Some properties of the LN distribution including moments, tail behavior, and modes are examined. The distribution is symmetric and can be unimodal or bimodal. The tail of the LN distribution can be heavier or lighter than the tail of the normal distribution. The performance of the maximum likelihood estimators is evaluated through small simulation study. Two bimodal data sets are used to show the applicability of the LN distributionItem Open Access On the well-posedness of the Boltzmann's moment system of equations in fourth approximation(Nazarbayev University School of Science and Technology, 2016-05) Issagali, AizhanWe study the one-dimensional non-linear non-stationary Boltzmann's moment system of equations in fourth approxi- mation with the tools developed by Sakabekov in [4],[5] and [6]. For the third approximation system Sakabekov proves the mass conservation law (cf. Theorem 2.1 in [4]) and discusses the existence and uniqueness of the solution (cf. Theorem in [6]). We extend the analysis of the existence and uniqueness of the solution to the fourth approximation system. In particular, for the fourth approximation system we discuss the well-posed initial and boundary value problem and obtain the a-priori estimate of the solution belonging to the space of functions, continuous in time and square summable by spatial variable.Item Open Access Optimization of convex geometries: component quadratic and general(Nazarbayev University School of Science and Technology, 2016) Myrzakul, ZhanbotaIn this Capstone Project, we worked with a class of closure systems called convex geometries, which are closure systems with a closure operator that satisfies the anti-exchange property. We first looked at the result of optimization algorithm of component quadratic systems, which are discussed in [4], and reproved it for the case of convex geometries. We then investigated the following question: if a convex geometry is given by a set of implications, is it possible to find its optimum basis in polynomial time when the convex geometry does not have particular properties (for instance, not component quadratic)?