###
Complex Analysis (MATH502)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

Recap of undergraduate complex analysis: holomorphic/analytic functions, Cauchy’s theorem, residue theorem. Conformal mappings, local degree. Möbius transformations. Techniques for evaluating complex integrals. Equivalent conditions for analyticity. The d-bar derivative. Plemelj and Pompeiu formulae. Weierstrass canonical product formula. Analytic continuation, and its uses in defining complex functions. Gamma, beta, and zeta functions. Jordan’s lemma. The inverse Laplace transform. Partial differential equations. The unified transform method.

###
Selected Topics in Probability Theory (MATH541)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Principal Component Analysis (MATH542)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 0 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in System Theory (MATH543)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in Algebraic Topology (MATH547)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in Differential Equations (MATH549)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Mathematical Statistics (MATH550)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 0 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in Analysis (MATH551)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** - |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in Analysis - II (MATH552)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Approximation Properties of Linear Positive Operators (MATH553)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Special Functions by Continued (MATH554)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Select Topics Differencial Geometry (MATH555)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Quantum Calculus (MATH556)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Function of Several Variables (MATH557)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in Applied Mathematics (MATH558)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Functional Analysis (MATH561)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Set Theory, Zorn's lemma. Topological spaces. Metric spaces. Linear spaces, Banach spaces, Hilbert spaces, Dual space. Completeness, separability, compactness. Linear operators and functionals, Riesz representation theorem. Hahn-Banach theorem. Contraction mapping. Strong and weak convergences.

###
Selected Topics in Functional Analysis (MATH563)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Special Functions (MATH564)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Fourier Analysis - I (MATH565)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 0 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Linear Algebra (MATH566)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Vector Spaces, Linear Transformations and Their Properties, Elementary Matrices, Determinants, Eigenvalues, Eigenvectors, Diagonalization, Inner Product Spaces; Gram- Schmidt Orthogonalization Process, Adjoint Forms, Brief Introduction to Canonical Forms.

###
Numerical Linear Algebra (MATH569)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Selected Topics in Numerical Analysis (MATH571)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Difference schemes for time-dependent equations with constant coefficients, Stability theory of difference schemes, Homogeneous difference schemes for time-dependent equations of mathematical physics, Economical difference schemes for multidimensional problems in mathematical physics are the main topics that will be considered in this course.

###
Computational Methods in PDE for Science and Engineering (MATH572)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Basic linear algebra. Method for designing difference schemes. Explicit, implicit formulae, convergence and stability for parabolic equations in one space dimension. ADI and LOD methods for parabolic eqautions in two space dimensions. ADI and LOD methods, the Neumann and Robbin’s problems for Laplace’s equation in a square. Laplace’s equation in three variables. Explicit, implicit and LOD methods for Hyperbolic equations.

###
Numerical Solutions of E.B.V.P. (MATH573)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Advanced Numerical Analysis (MATH574)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 0 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Fractional Calculus (MATH576)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 0 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Special functions for fractional calculus: gamma, beta, hypergeometric, Mittag-Leffler functions. The Riemann-Liouville model of fractional calculus, with properties including semigroup law, differentiation of series, Leibniz rule, and Laplace transforms. The Caputo, Weyl, Cauchy, and Grunwald-Letnikov models of fractional calculus, and their relationships with the Riemann-Liouville model. Some simple ordinary differential equations solved using transform methods. Fractional calculus defined using special-function kernels; basic properties of these models and their relationships with the Riemann-Liouville model.

###
Fractional Differential Equations (MATH577)

**Credit:** 0 |

**Lecture Hour (hrs/week):** 0 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Banach Fixed Point Theory, Schauder Fixed Point Theorem, Fractional Differentiation and Integration, Application of Fixed Point Theory, Existence and Uniqueness of Solutions of various types of Fractional Differential Equations.

###
Theory of Finite Difference Schemes (MATH578)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Block Method for solving the Laplace equation (MATH580)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 0 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Real Analysis (MATH583)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Review on Riemann integral. Infinite sets. Countable and continuum sets. Equivalent sets. Cardinality. Point set topology. Systems of sets. Algebras, sigma-algebras and semi-algebras. Monotonic class theorem. Measure and measurable sets. Construction of measures. Caratherodory theorem. Hahn theorem. Lebesgue and Stieltjes measures. Measurable functions. Lebesgue Iitegral and its properties. Comparison of Lebesgue and Riemann integrals. The spaces of Lebesgue integrable functions. Product measure. Tonelli and Fubini theorems. Absolute continuity. Radon-Nikodym theorem.

###
Differential Geometry (MATH584)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Theory of Optimal Control (MATH585)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Recent Developments in Computational Fluid Dynamics (MATH586)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Advanced Engineering Mathematics (MATH587)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 0 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

Fourier series, complex Fourier series, Approximation by trigonometric polynomials, Fourier integral, Fourier transforms, discrete and fast Fourier transforms, wave equation, solution by separating variables, D’Alembert solution of the wave equation, characteristics, heat equation: solution by Fourier series, solution by Fourier integrals and transforms, two-dimensional wave equation, rectangular membrane, double Fourier series, Laplacian in polar coordinates, circular membrane, Fourier-Bessel series, potentials, solutions of PDEs by Laplace transforms.

###
Numerical Sol. of Parabolic PDE and Inverse Control Prob. (MATH588)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Linear Programming (MATH589)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Nonlinear Differential Equations and Dynamic Systems (MATH590)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Dynamical Systems: Introduction to Bifurcation (MATH591)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Introduction to Nonlinear Programming (MATH592)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Introduction to Bifurcation II (MATH593)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Problem solving in Analysis, Algebra, and Topology (MATH597)

**Credit:** 0 |
**Lecture Hour (hrs/week):** 0 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Mathematical Achievements In Antiquity (MATH599)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Korovkin Type Approximation Theory (MATH653)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Advanced Numerical Linear Algebra (MATH669)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

This course aims to study iterative solution methods for solvıng linear systems mainly using preconditioning methods. As a Krylov subspace iterative method accelerated over relaxation method and the convergence theorems of this method for L- matrices, ırreducible matrices, consistently ordered matrices will be given. Preconditioning of linear systems will be studied by incomplete factorization preconditioning methods, approximate matrix inverses and corresponding preconditioning methods, block diagonal and Schur complement preconditionings. Estimates of eigenvalues and condition numbers for preconditioned matrices will be analyzed .

###
Nabla Fractional Calculus (MATH670)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 1 |
**ECTS:** -

###
Orthogonal Polynomials On Non Uniform Lattices (MATH672)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

In this course the q-Analogs of the Hahn, Meixner, Kravchuk,and Charlier Polynomials on the lattices x(s) = exp(2ws) and x(s) = sinh(2ws) will be investigated. Also, the q-Analogs of the Racah and Dual Hahn polynomials on the lattices x(s) = cosh(2ws) and x(s) = cos(2ws) will be considered. Asymptotic properties of the Racah and Dual Hahn polynomials and construction of some orthogonal polynomials on non-uniform lattices by means of the Darboux-Christoffel formula will be studied.

###
Fractional Abstract Evolution Equations (MATH677)

**Credit:** 3 |
**Lecture Hour (hrs/week):** 3 |
**Lab (hrs/week):** 0 |
**Tutorial (hrs/week): ** 0 |
**ECTS:** -

###
Advanced methods of Applied Mathematics (MATH687)

**Credit:** 3 |

**Lecture Hour (hrs/week):** 3 |

**Lab (hrs/week):** 0 |

**Tutorial (hrs/week): ** 0 |

**ECTS:** -

The Sturm Liouville theory with engineering applications such as elastic vibration, buckling of beams and vibration of circular membrane will be studied. Finite Difference numerical methods that are based on the principle of discretization which results in generated numerical values of the solution at a discrete set of points will be investigated. Finally after investigating the structure of Green’s function, the solutions of Dirichlet’s and Neumann problems will be given with the help of Conformal mappings.