The Department of Mathematicsand Computer Scienceoffers two graduate programs leading to the degree of Master of Science in Mathematics.The objective of the program is to provide courses in mathematics to graduates of arts and sciences and engineering who may already have jobs, and who aim to gain further education in mathematics.

**The following prerequisites apply for the practice and completion of the Master of Science Degrees in Mathematics with ThesisProgram:**

**The time to complete the program is maximumthree years and minimum three semesters.**Candidates who cannot finish on time can still register in case they pay tuition fee or studentship fee in conjunction with the Act of Law 2547, clause 46. In such a case, the studentship is limited to participation to the classes and preparation of the thesis and all other rights cease.- For students who successfully complete their for-credit courses and Seminar Course but are unable to enter the thesis exam because they could not finish their thesis by the end of the fourth semester, with the recommendation of the Department of Mathematics and Computer Science and the approval of the Institute Administrative Board an additional time of at most two semesters can be provided to take the thesis exam and defend his or her thesis.
**Course Requirements:****A student must take minimumtwenty one credits in total with seven courses in addition to MCS 590****Graduate Seminar, MCS 591 Special Studies and MCS 599 Thesis**Three of these courses must be chosen from the**Compulsory Course Set**and the rest from the**Elective Course Set**provided in the table below and/or from other departments with the approval of their advisor.**Upon completing these courses**, students who wish may also take additional courses.For students who take courses in addition to their minimum course load with the recommendation of the Department of Mathematics and Computer Scienceand the approval of the Institute Administrative Board, their grades in these courses will not be factored into their grade point average but will be included in their transcript.- Seminar and Thesis courses: Students must take their Seminar, Special Studies and Thesis courses at the latest from third semester.

The** Department of Mathematics and Computer Science will recommend a thesis topic and thesis advisor for each student to the pertaining institute latest by the end of the second semester.** The topic and advisor recommendation will be finalized with the approval of the Institute Administrative Board. In the event that the nature of the thesis requires more than one thesis advisor, a second thesis advisor may be designated with the recommendation of the Department of Mathematics and Computer Science and the approval of theInstitute Administrative Board.

**Additionally**, as stated in the Çankaya University Graduate Education Regulationsin the Graduate Program withThesis,- A student may take
**undergraduate**courses with the recommendation of the Department of Mathematics and Computer Science and the approval of the Institute Administrative Board. However,**only two**of these courses at the most will be factored in when calculating the course load and graduate degree credits. - The student may take graduate courses at
**universities other than ÇankayaUniversity**to be factored into the course load with the recommendation of the Academic Committee of the Department of Mathematics and Computer Science and the approval of the Institute Administrative Board. The number of these courses**cannot exceed three**.

- A student may take

Compulsory Courses |
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Course Code |
Title |
Credit |
Course Content |

MCS 501 | Analysis | 3 0 3 | Elementary topology of Rn, continuous functions in Rn, uniform continuity, uniform convergence, differentiability and implicit function theorem, differentiation under an integral sign, Stone-Weierstrass theorem on the real line, measure spaces, Lebesgue measure and integral, convergence theorems for the Lebesgue integral, types of convergence for sequences of functions, product measures and Fubini’s theorem, Lp spaces and the Riesz representation theorem, Radon-Nikodym theorem. |

MCS 502 | Ordinary Differential Equations | 3 0 3 | Basic theory: initial value problems. Linear systems: linear homogeneous and non homogeneous systems. Linear systems with constant and periodic coefficients. Oscillation theory. Stability: definitions of stability and its boundedness. Lyapunov functions. Lyapunov stability and instability. Domain of attraction. Perturbation of linear systems. Stability of an equilibrium point. The stable manifold. Stability of periodic solutions. Asymptotic equivalence. |

MCS 506 | Algebra | 3 0 3 | Groups: generalities, groups acting on a set, Sylow theorems, free group, direct product and sums. Rings: generalities, commutative rings, principle ideal domains, unique factorization domains, Euclidean domains. Noetherian rings. Hilbert’s theorem. Field of fractions. Localization. |

Elective Courses (Four of the Following Courses) |
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Course Code |
Title |
Credit |
Course Content |

MCS 503 | Scientific Computation I | 3 0 3 | Gaussian elimination and its variants. Sensitivity of linear systems. Orthogonal matrices and the least squares problem. Eigenvalues and eigenvectors. The singular value decomposition. Solutions of Partial differential equations, solution of system of equations, Examples of time dependent events, and their solutions. Applications with MATLAB / Java. |

MCS 507 | Partial Differential Equations | 3 0 3 | Cauchy-Kowalevski theorem. Linear and quasilinear first order equations. Existence and uniqueness theorems for second order elliptic, parabolic and hyperbolic equations. Correctly posed problems. Green’s function. |

MCS 510 | Applied Functional Analysis | 3 0 3 | Distribution theory and Green’s functions, the Delta function, basic distribution theory, convergence of distributions, The integral of a distribution, Applications of Green’s functions, The classical Fourier transform, Distributions of slow growth, generalized Fourier transforms, Banach spaces and fixed point theorems, the contraction mapping theorem, Application to differential and integral equations, Hilbert spaces, orthogonal expansions, bounded operators on normed spaces, eigenvalue problems for self-adjoint operators, variational methods, positive operators, the Rayleigh- Ritz method for eigenvalues, applications. |

MCS 512 | Scientific Computation II | 3 0 3 | Interpolation: Polynomial interpolation, Divided differences, Hermite interpolation, Spline interpolation. Approximation of functions. Numerical differentiation: Richardson extrapolation. Numerical integrations: Guassian Quadrature, Romberg integration. Root finding methods:Bisection, Newton, Secant methods, Fixed point iteration. Applications with MATLAB. |

MCS 513 | Nonlinear Dynamical Systems | 3 0 3 | Equilibrium solutions, Lyapunov Functions, Periodic Solutions, Poincare maps, center manifolds, normal forms, bifurcation. |

MCS 514 | Special Topics in Fractional Differential Equations | 3 0 3 | Fractional integrals and derivatives, Cauchy type problem for ordinary fractional linear equations, Fractional existence and uniqueness theorems, Fractional method of reduction to fractional Volterra integral equations, Fractional compositional method. Applications with MATLAB. |

MCS 515 | Special Topics in Applied Convex Functions | 3 0 3 | Convex functions on Intervals, the integral form of Jensen’s inequality, the Hermite-Hadamard Inequality, convexity and majorization, Comparative Convexity on Intervals, the Gamma and Beta functions, Multiplicative convexity of special functions, Convex functions on Banach spaces, Continuity, Differentiability of convex functions, the variational approach of Partial Differential Equations, the minimum of convex functionals. |

MCS 516 | Spectral Theory of Linear Operators | 3 0 3 | Compact operators, compact operators in Hilbert spaces, Banach Algebras, The spectral theorem of normal operators, unbounded operators between Hilbert spaces, The spectral theorem for unbounded adjoint operators, self-adjoint operators, self adjoint extentions. |

MCS 517 | Advanced Dynamic Equations On Time Scales | 3 0 3 | Linear Systems, Initial Value Problems, Existence and Uniqueness of Solutions, Self-Adjoin Matrix Equatios, Asymptotic Bahavior of Solutions, Oscillation Theory, Higher Order Linear Dynamic Equations, Dynamic Inequalities, Upper and Lower Solitions, Linear Symplectic Dinamic Systems, Nonlinear Theory |

MCS 519 | Difference Equations | 3 0 3 | The difference calculus, first order equations, linear equations, equations with constant coefficients, equations with variable coefficients, undetermined coefficients method, variation of parameters method, the Z-transform, linear systems, stability theory |

At the latest from Third Semester |
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MCS 590 | Graduate Seminar | – |

MCS 591 | Special Studies | 0 4 0 |

MCS 599 | Thesis | – |

For more information,**Çankaya University Graduate Education**** Regulations**.

Please click here to see **Sample Scenario.**

Sample Scenario |
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1st Semester |
2nd Semester |
3rd Semester |
4th Semester |
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Compulsory Course I | Compulsory Course III | Special Studies | Special Studies | |||

Compulsory Course II | Elective Course III | Thesis | Thesis | |||

Elective Course I | Elective Course IV | |||||

Elective Course II | Graduate Seminar |