# Calculus on Time Scales (MATH363) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Calculus on Time Scales MATH363 Area Elective 3 0 0 3 6
Pre-requisite Course(s)
MATH 136
Course Language English Elective Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Discussion. A time scale is an arbitrary nonempty closed subset of the real numbers. The calculus of time scales was initiated by B.Aulbach and S.Hilger in 1990 in order to create a theory that can unify discrete and continuous analysis. It allows a simultaneous treatment of differential and difference equations, extending those theories to so-called dynamic equations. This course gives an introduction to this subject. The students who succeeded in this course; perform differentiation and integration of functions whose domain of definition may have more complicated structure containing discrete and continuous parts, understand and apply the special functions on time scales, have tools for mathematical modeling of practical problems having discrete and continuous components simultaneously. The h-derivative and the q-derivative, the concept of a time scale, differentiation on time scales, integration on time scales, Taylor?s Formula on time scales.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Rules of the h-differentiation Kac and Cheung, pp. 80-84
2 Rules of the q –differentiation Kac and Cheung, pp. 1-3
3 Definition of a time scale, examples pp. 1-4
4 Delta and nabla derivatives, Differentiation rules pp. 5- 21, 331-332
5 Chain rules on time scales pp. 31-37
6 Mean value theorems for delta and nabla derivatives pp. 22-26
7 Midterm
8 The Riemann delta and nabla integrals on time scales pp. 26-27, 332-333
9 Properties of the integral, Integration by parts formulas pp. 28-30
10 The Fundamental Theorems of Caculus on time scales p. 27
11 Mean Value Theorems for integral on time scales pp.48-49
12 Improper integrals on time scales pp. 30-31
13 Generalized monomials on time scales pp. 37-42
14 Taylor’s formula on time scales pp. 42-46
15 h-Taylor’s formula and q-Taylor’s formula Kac and Cheung, pp. 7-13
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 X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 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 X
4 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 X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 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 X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
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. X

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 8 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 15 30
Prepration of Final Exams/Final Jury