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| MAT7200 | Spectral Theory of Periodic Diff. Eq. | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Third Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | Face to face | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Prof. Dr. Haskız COŞKUN | | Co-Lecturer | None | | Language of instruction | | | Professional practise ( internship ) | None | | | | The aim of the course: | | Some periodic differential equations,stability theory, eigenvalues and eigenfunctions of periodic differential equations. |
| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | learn about Floquet theory. | 1,2,3,4 | 1 | | PO - 2 : | understand the second order periodic differential equation with real periodic coefficients commonly known as Hill's equation. | 1,2,3,4 | 1 | | PO - 3 : | learn about stability theory and eigenfunction expansion formulae. | 1,2,3,4 | 1 | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| Stability and instability intervals. Hill equation. Boundness of periodic solutions. Complex valued solutions. Periodic and semi-periodic problem. t- periodic eigenvalue problem. Mathieu equation. Comparison of eigenvalues of different eigenvalue problems. Prüfer mapping formulas. The Liouville transformation and its applications. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Floquet theory,Hill's equation | | | Week 2 | Boundedness and periodicity of solutions,complex valued coefficients | | | Week 3 | Systems of differential equations,systems all of whose solutions are periodic | | | Week 4 | İntroduction to stability and instability intervals | | | Week 5 | The periodic and semi-periodic eigenvalue problems | | | Week 6 | The discriminant function,two further eigenvalue problems | | | Week 7 | The Mathieu equation | | | Week 8 | Zeros of eigenfunctions,oscillation of solutions,two inequalities concerning periodic eigenvalues | | | Week 9 | Mid-term exam | | | Week 10 | The right-hand end-points of the instability intervals,stability criteria | | | Week 11 | Prüfer transformation formulae,asymptotic estimates | | | Week 12 | Asymptotic formulae for solutions,an improvement of the formulae | | | Week 13 | The length of the instability intervals | | | Week 14 | İntroduction to inverse problems | | | Week 15 | Some important results in the inverse problems | | | Week 16 | End-of-term exam | | | |
| 1 | Eastham,M.S.P,1973,The Spectral Theory of Periodic differenial equations,Scottish Academic Press,Edinburgh,127pp | | | |
| 1 | Zettl,Anton,2005,Sturm-Liouville Theory,American Mathematical Society,USA,328 pp | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 8 | 31/03/2016 | 2 | 30 | | In-term studies (second mid-term exam) | 12 | 21/04/2016 | 2 | 20 | | End-of-term exam | 16 | 12/05/2016 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 3 | 14 | 42 | | Sınıf dışı çalışma | 6 | 14 | 84 | | Arasınav için hazırlık | 15 | 1 | 15 | | Arasınav | 2 | 1 | 2 | | Ödev | 5 | 3 | 15 | | Kısa sınav | 2 | 1 | 2 | | Dönem sonu sınavı için hazırlık | 20 | 1 | 20 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 182 |
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