ECTS - Applied Mathematics
Applied Mathematics (MATH587) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Applied Mathematics | MATH587 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer, Problem Solving. |
| Course Lecturer(s) |
|
| Course Objectives | The course is divided into two parts: Calculus of Variations and Integral Equations. In the first part, the course aims to present the main elements of the calculus of variations. The approach is oriented towards the differential equation aspects. We will focus on variational problems that involve one and two independent variables. The fixed end-point problem and problems with constraints will be discussed in detail. Topics will also include Euler-Lagrange equation, the first and second variations, necessary and suffcient conditions for extrema, Hamilton's principle, and application to Sturm-Liouville problems and mechanics. In the second part, the course aims to introduce student the integral equations and their connections with initial and boundary value problems of differential equations. Topics will include mainly Fredholm and Volterra integral equations, the Green’s function, Hilbert-Schmidt theory, the Neumann series and Fredholm theory. |
| Course Learning Outcomes |
The students who succeeded in this course;
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| Course Content | Calculus of variations: Euler-Lagrange equation, the first and second variations, necessary and sufficient conditions for extrema, Hamilton`s principle, and applications to Sturm-Liouville problems and mechanics; integral equations: Fredholm and Volterra integral equations, the Green?s function, Hilbert-Schmidt theory, the Neumann series and Fredho |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | Calculus of Variations and Applications:Maxima and minima of one variable and multivariable functions.The subject of calculus of variations. | |
| 2 | The simplest case of variational problems. Necessary condition for the existence of an extremum: the Euler equation. Extremals. | |
| 3 | Natural boundary conditions and transition conditions. Function spaces and functionals. | |
| 4 | The concept of variation of functionals. A case of integrals depending on functions of two variables. | |
| 5 | The more general case of variational problems. Variational problems with variable endpoints. | |
| 6 | Application to Sturm-Liouville problems. Application to mechanics: Hamilton’s principle, Langrange’s equations, Hamilton’s canonical equations. | |
| 7 | Basic Definitions. Fredholm and Volterra integral equations. | |
| 8 | Midterm Exam | |
| 9 | Relations between differential and integral equations. | |
| 10 | The Green’s function. | |
| 11 | Fredholm equations with separable kernels. | |
| 12 | Hilbert-Schimidt theory. | |
| 13 | Iterative methods for solving an integral equation of second kind. The Neumann series. | |
| 14 | Fredholm theory.Singular integral equations. Special devices for solving some integral equations. | |
| 15 | Methods for obtaining approximate solutions of integral equations. | |
| 16 | Final Exam |
Sources
| Course Book | 1. F. B. Hildebrand, Methods of Applied Mathematics, 2nd Edition, 1965, Prentice – Hall, Englewood Cliffs. |
|---|---|
| Other Sources | 2. I. M. Gelfand and S. V. Fomin, Calculus of Variations, 1963, Prentice – Hall, Englewood Cliffs. |
| 3. W. V. Lovitt, Linear Integral Equations, 1924, McGraw – Hill, New York. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 5 | 30 |
| Presentation | - | - |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 30 |
| Final Exam/Final Jury | 1 | 40 |
| Toplam | 7 | 100 |
| Percentage of Semester Work | 60 |
|---|---|
| Percentage of Final Work | 40 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Ability to carry out advanced research activities, both individual and as a member of a team | |||||
| 2 | Ability to evaluate research topics and comment with scientific reasoning | |||||
| 3 | Ability to initiate and create new methodologies, implement them on novel research areas and topics | |||||
| 4 | Ability to produce experimental and/or analytical data in systematic manner, discuss and evaluate data to lead scintific conclusions | |||||
| 5 | Ability to apply scientific philosophy on analysis, modelling and design of engineering systems | |||||
| 6 | Ability to synthesis available knowledge on his/her domain to initiate, to carry, complete and present novel research at international level | |||||
| 7 | Contribute scientific and technological advancements on engineering domain of his/her interest area | |||||
| 8 | Contribute industrial and scientific advancements to improve the society through research activities | |||||
ECTS/Workload Table
| 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 | 14 | 3 | 42 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| Homework Assignments | 5 | 3 | 15 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 10 | 10 |
| Prepration of Final Exams/Final Jury | 1 | 10 | 10 |
| Total Workload | 77 | ||