We consider the reaction-diffusion-advection (RDA) equations resulting in air pollution mod- eling problems. We employ the finite element method (FEM) for solving the RDA equations in two dimensions. Linear triangular finite elements are used in the discretization of problem domains. The instabilities occuring in the solution when the standard Galerkin finite element method is used, in advection or reaction dominated cases, are eliminated by using an adap- tive stabilized finite element method. In transient problems the unconditionally stable Crank- Nicolson scheme is used for the temporal discretization. The stabilization is also applied for reaction or advection dominant case in the time dependent problems. It is found that the stabilization in FEM makes it possible to solve RDA problems for very small diffusivity constants. However, for transient RDA problems, although the stabilization improves the solution for the case of reaction or advection dominance, it is not that pronounced as in the steady problems. Numerical results are presented in terms of graphics for some test steady and unsteady RDA problems. Solution of an air pollution model problem is also provided.

Regulatory processes and history dependent behavior appear in many dynamical systems in nature and technology. For modeling regulatory processes, hybrid systems offer several advances. From this point of view, to observe the capability of hybrid systems in a history dependent system is a strong motivation. In this thesis, we developed functional hybrid systems which exhibit memory dependent behavior such that the dynamics of the system is determined by both the location of the state vector and the memory. This property was explained by various examples. We used the hybrid system with memory in modeling the gene regulatory network of human immune response to Influenza A virus infection. We investigated the sensitivity of the piecewise linear model with memory. We introduced how the model can be developed in future.

In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.

Semi-infinite programming problems is a class of optimization problems in finite dimensional variables which are subject to infinitely many inequality constraints. If the infinite index of inequality constraints depends on the decision variable, then the problem is called generalized semi-infinite programming problem (GSIP). If the infinite index set is fixed, then the problem is called standard semi-infinite programming problem (SIP). In this thesis, convergence of a semismooth Newton method for generalized semi-infinite programming problems with convex lower level problems is investigated. In this method, using nonlinear complementarity problem functions the upper and lower level Karush-Kuhn-Tucker conditions of the optimization problem are reformulated as a semismooth system of equations. A possible violation of strict complementary slackness causes nonsmoothness. In this study, we show that the standard regularity condition for convergence of the semismooth Newton method is satisfied under natural assumptions for semi-infinite programs. In fact, under the Reduction Ansatz in the lower level problem and strong stability in the reduced upper level problem this regularity condition is satisfied. In particular, we do not have to assume strict complementary slackness in the upper level. Furthermore, in this thesis we neither assume strict complementary slackness in the upper nor in the lower level. In the case of violation of strict complementary slackness in the lower level, the auxiliary functions of the locally reduced problem are not necessarily twice continuously differentiable. But still, we can show that a standard regularity condition for quadratic convergence of the semismooth Newton method holds under a natural assumption for semi-infinite programs. Numerical examples from, among others, design centering and robust optimization illustrate the performance of the method.

Analysis of large networks in biology, science, technology and social systems have become very popular recently. These networks are mathematically represented as graphs. The task is then to extract relevant qualitative information about the empirical networks from the analysis of these graphs. It was found that a graph can be conveniently represented by the spectrum of a suitable difference operator, the normalized graph Laplacian, which underlies diffusions and random walks on graphs. When applied to large networks, this requires computation of the spectrum of large matrices. The normalized Laplacian matrices representing large networks are usually sparse and unstructured. The thesis consists in a systematic evaluation of the available eigenvalue solvers for nonsymmetric large normalized Laplacian matrices describing directed graphs of empirical networks. The methods include several Krylov subspace algorithms like implicitly restarted Arnoldi method, Krylov-Schur method and Jacobi-Davidson methods which are freely available as standard packages written in MATLAB or SLEPc, in the library written C++. The normalized graph Laplacian as employed here is normalized such that its spectrum is confined to the range [0, 2]. The eigenvalue distribution plays an important role in network analysis. The numerical task is then to determine the whole spectrum with appropriate eigenvalue solvers. A comparison of the existing eigenvalue solvers is done with Paley digraphs with known eigenvalues and for citation networks in sizes 400, 1100 and 4500 by computing the residuals.

Regression analysis refers to techniques for modeling and analyzing several variables in statistical learning. There are various types of regression models. In our study, we analyzed Generalized Partial Linear Models (GPLMs), which decomposes input variables into two sets, and additively combines classical linear models with nonlinear model part. By separating linear models from nonlinear ones, an inverse problem method Tikhonov regularization was applied for the nonlinear submodels separately, within the entire GPLM. Such a particular representation of submodels provides both a better accuracy and a better stability (regularity) under noise in the data. We aim to smooth the nonparametric part of GPLM by using a modified form of Multiple Adaptive Regression Spline (MARS) which is very useful for high-dimensional problems and does not impose any specific relationship between the predictor and dependent variables. Instead, it can estimate the contribution of the basis functions so that both the additive and interaction effects of the predictors are allowed to determine the dependent variable. The MARS algorithm has two steps: the forward and backward stepwise algorithms. In the rst one, the model is built by adding basis functions until a maximum level of complexity is reached. On the other hand, the backward stepwise algorithm starts with removing the least significant basis functions from the model. In this study, we propose to use a penalized residual sum of squares (PRSS) instead of the backward stepwise algorithm and construct PRSS for MARS as a Tikhonov regularization problem. Besides, we provide numeric example with two data sets / one has interaction and the other one does not have. As well as studying the regularization of the nonparametric part, we also mention theoretically the regularization of the parametric part. Furthermore, we make a comparison between Infinite Kernel Learning (IKL) and Tikhonov regularization by using two data sets, with the difference consisting in the (non-)homogeneity of the data set. The thesis concludes with an outlook on future research.

This study focuses on the effects of different constitutive models in sheet metal forming operations by considering the cylindrical and square cup drawing and V-bending simulations. Simulations are performed using eight different constitutive models / elastic plastic constitutive model with isotropic hardening, elastic plastic constitutive model with kinematic hardening, elastic plastic constitutive model with combined hardening, power law isotropic plasticity, piecewise linear isotropic plasticity, Barlatthree-parameter, cyclic elastoplastic and Hill&rsquo / 48 model.The numerical analyses are accomplished by using three different 1 mm thick sheet materials / St12 steel, Al-5182 aluminum and stainless steel 409 Ni. An explicit finite element code is used in the simulations. For square cup drawing, three different blank holder forces / 2 kN, 4 kN and 5 kN are considered for St12 steel, whereas only 5 kN blank holder force is applied for stainless steel 409 Ni and Al-5182 aluminum. A number of experiments are carried out and analytical calculations are utilized to evaluate the results of simulations. In cylindrical cup drawing, simulation results of different constitutive models show good agreement with analytical calculations for thickness strain and effective stress distributions. In square cup drawing, simulation results of all the models displayed good agreement with the experimental results for edge contour comparisons, although the distributions of effective stress vary for different models within the cup. The numerically and experimentally obtained springback amounts are also in good agreement. The simulation results obtained for piecewise linear isotropic plasticity and power law isotropic plasticity models show better agreement with the analytical solutions and experiments.

In this thesis, we consider a projection-based stabilization method for solving buoyancy driven flows (natural convection problems). The method consists of adding global stabilization for all scales and then anti-diffusing these effects on the large scales defined by projections into appropriate function spaces. In this way, stabilization acts only on the small scales. We consider two different variations of buoyancy driven flows based on the projection-based stabilization. First, we focus on the steady-state natural convection problem of heat transport through combined solid and fluid media in a classical enclosure. We present the mathematical analysis of the projection-based method and prove existence, uniqueness and convergence of the approximate solutions of the velocity, temperature and pressure. We also present some numerical tests to support theoretical findings. Second, we consider a system of combined heat and mass transfer in a porous medium due to the natural convection. For the semi-discrete problem, a stability analysis of the projectionbased method and a priori error estimate are given for the Darcy-Brinkman equations in double-diffusive convection. Then we provide numerical assessments and a comparison with some benchmark data for the Darcy-Brinkman equations. In the last part of the thesis, we present a fully discrete scheme with the linear extrapolation of convecting velocity terms for the Darcy-Brinkman equations.

Inferring and anticipation of genetic networks based on experimental data and environmental measurements is a challenging research problem of mathematical modeling. In this thesis, we discuss gene-environment network models whose dynamics are represented by a class of time-continuous systems of ordinary differential equations containing unknown parameters to be optimized. Accordingly, time-discrete version of that model class is studied and improved by using different numerical methods. In this aspect, 3rd-order Heun&rsquo / s method and 4th-order classical Runge-Kutta method are newly introduced, iteration formulas are derived and corresponding matrix algebras are newly obtained. We use nonlinear mixed-integer programming for the parameter estimation and present the solution of a constrained and regularized given mixed-integer problem. By using this solution and applying the 3rd-order Heun&rsquo / s and 4th-order classical Runge-Kutta methods in the timediscretized model, we generate corresponding time-series of gene-expressions by this thesis. Two illustrative numerical examples are studied newly with an artificial data set and a realworld data set which expresses a real phenomenon. All the obtained approximate results are compared to see the goodness of the new schemes. Different step-size analysis and sensitivity tests are also investigated to obtain more accurate and stable predictions of time-series results for a better service in the real-world application areas. The presented time-continuous and time-discrete dynamical models are identified based on given data, and studied by means of an analytical theory and stability theories of rarefication, regularization and robustification.

The globalization leads to integration of the economies worldwide. As the firms&#039 / businesses also get integrated with each other, the financing choices of the firms diversify. Among these choices, the popularity and the share of foreign currency borrowing in total borrowing by non-financial firms increase in Turkey similar to the global developments. The main purpose of this thesis is to price the risks of default and financial distress due to foreign currency denominated loans of non-financial firms in Turkey. The valuation model of foreign currency corporate loans is established by two state variable option pricing model based on the study of Cox, Ingersoll and Ross. In our model, the main risk factors are identified as the exchange rate and the interest rate, which are the state variables of the main partial differential equation whose solution gives the value of the asset. The numerical results are tested for different parameters and for different economic environments. The findings show that interest rate fluctuations are more important both for the default and financial distress option values than the fluctuations in exchange rate. However, the effect of upside movements of exchange rate on the financial distress and default values is sharper than the downside movement effect of interest rate. Furthermore, high loan-to-value (LTV) foreign currency loans result in significantly high financial distress values that cannot be disregarded and can lead to default of the firm. To the best of our knowledge, this thesis is the first study that develops a structural model to evaluate foreign currency denominated corporate loans in an option-pricing framework.