# Dynamic Systems on Time Scales (MATH565) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Dynamic Systems on Time Scales MATH565 3 0 0 3 5
Pre-requisite Course(s)
Consent of the instructor
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Lecture, Question and Answer, Team/Group. Bu ders özellikle matematik, fizik ve mühendislik bölümü öğrencilerinden diskrit (fark) ve sürekli (diferensiyel) denklemlerin birleştirilmesinden ortaya çıkan melez denklemleri kullanan öğrencilere hitap etmektedir. Bu amaçla bu ders zaman skalasında diferensiyel denklemleri sunarak onların çözim yöntemlerini verecektir. The students who succeeded in this course; understand and apply the differentiation and integration on time scales, understand the basic properties of differential equations (dynamic systems) on time scales and know methods of their solution, know the basic results related to the Sturm-Liouville eigenvalue problem on time scales. Differentiation on time scales, integration on time scales, the first-order linear differential equations on time scales, initial value problem, the exponential function on time scales, the second-order linear differential equations on time scales, boundary value problem, Green?s function, the Sturm-Liouville eigenvalue problem.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Differentiation on time scales. pp. 5-21
2 Integration on time scales. pp. 22-31
3 The existence and uniqueness theorem for solution of the initial value problem for first-order differential equations on time scales. pp. 321-326
4 Definition of the exponential function on time scales via a differential equation and properties of the exponential function. pp. 58-68
5 Examples of exponential functions on time scales. pp. 69-74
6 Solution of the first-order linear differential equations with variable coefficients on time scales. pp. 75-78
7 Midterm
8 The second-order linear homogeneous differential equations on time scales, The Wronskian. pp. 81-87
9 Definitions of the cosine and sine on time scales, and their properties. pp. 87-93
10 Solving of the second order linear differential equations with constant coefficients on time scales. pp. 93-96
11 The second order linear nonhomogeneous differential equations on time scales, Variations of parameters. pp. 113-116
12 Boundary value problems for second-order linear differential equations on time scales, The Green function. pp. 164-177
13 The Sturm-Liouville eigenvalue problem on time scales. pp. 177-183
14 Expansion formulas in eigenfunctions on time scales. pp. 183-187
15 Higher-order linear differential equations on time scales. pp. 238-253
16 Final Exam

### Sources

Course Book 1. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. 2. V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002. 3. V. Lakshimikantham, S Sivasundaram, and B. Kaymakçalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. 4. M. Bohner and A. Peterson, editors, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 40
Toplam 8 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 2 10
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 7 14
Prepration of Final Exams/Final Jury 1 11 11