# 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 Accumulated knowledge on mathematics, science and mechatronics engineering; an ability to apply the theoretical and applied knowledge of mathematics, science and mechatronics engineering to model and analyze mechatronics engineering problems. X
2 An ability to differentiate, identify, formulate, and solve complex engineering problems; an ability to select and implement proper analysis, modeling and implementation techniques for the identified engineering problems. X
3 An ability to design a complex system, product, component or process to meet the requirements under realistic constraints and conditions; an ability to apply contemporary design methodologies; an ability to implement effective engineering creativity techniques in mechatronics engineering. (Realistic constraints and conditions may include economics, environment, sustainability, producibility, ethics, human health, social and political problems.) X
4 An ability to develop, select and use modern techniques, skills and tools for application of mechatronics engineering and robot technologies; an ability to use information and communications technologies effectively.
5 An ability to design experiments, perform experiments, collect and analyze data and assess the results for investigated problems on mechatronics engineering and robot technologies.
6 An ability to work effectively on single disciplinary and multi-disciplinary teams; an ability for individual work; ability to communicate and collaborate/cooperate effectively with other disciplines and scientific/engineering domains or working areas, ability to work with other disciplines.
7 An ability to express creative and original concepts and ideas effectively in Turkish and English language, oral and written.
8 An ability to reach information on different subjects required by the wide spectrum of applications of mechatronics engineering, criticize, assess and improve the knowledge-base; consciousness on the necessity of improvement and sustainability as a result of life-long learning; monitoring the developments on science and technology; awareness on entrepreneurship, innovative and sustainable development and ability for continuous renovation.
9 Be conscious on professional and ethical responsibility, competency on improving professional consciousness and contributing to the improvement of profession itself.
10 A knowledge on the applications at business life such as project management, risk management and change management and competency on planning, managing and leadership activities on the development of capabilities of workers who are under his/her responsibility working around a project.
11 Knowledge about the global, societal and individual effects of mechatronics engineering applications on the human health, environment and security and cultural values and problems of the era; consciousness on these issues; awareness of legal results of engineering solutions.
12 Competency on defining, analyzing and surveying databases and other sources, proposing solutions based on research work and scientific results and communicate and publish numerical and conceptual solutions.
13 Consciousness on the environment and social responsibility, competencies on observation, improvement and modify and implementation of projects for the society and social relations and be an individual within the society in such a way that planing, improving or changing the norms with a criticism.
14 A competency on developing strategy, policy and application plans on the mechatronics engineering and evaluating the results in the context of qualitative processes.

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