| Course Name |
Integral Equations
|
|
Code
|
Semester
|
Theory
(hour/week) |
Application/Lab
(hour/week) |
Local Credits
|
ECTS
|
|
MATH 666
|
Fall/Spring
|
3
|
0
|
3
|
7.5
|
| Prerequisites |
None
|
|||||
| Course Language |
English
|
|||||
| Course Type |
Elective
|
|||||
| Course Level |
Third Cycle
|
|||||
| Mode of Delivery | - | |||||
| Teaching Methods and Techniques of the Course | - | |||||
| National Occupation Classification | - | |||||
| Course Coordinator | - | |||||
| Course Lecturer(s) | ||||||
| Assistant(s) | - | |||||
| Course Objectives | Integral Equations course, is theoretic course which builds bridges between functional analysis theory and applied mathematics theory. In this course, fundamental theory of integral equations are provided by establishing a connection between differential equations and integral equations. The fundamental theories of functional analysis is applied to integral equations and is illustrated in applied sciences. |
| Learning Outcomes |
The students who succeeded in this course;
|
| Course Description | Classification of Integral Equations, Connection with differential equations, kernels, special types of kernels, Fredholm and Volterra equations of the first and second kind, Method of successive approximations, Integral equations with singular kernels, Hilbert space, kernel and transforms. Linear integral operators in Hilbert spaces. The resolvent, Fredholm theory, degenerate kernels. |
| Related Sustainable Development Goals |
|
|
|
Core Courses | |
| Major Area Courses |
X
|
|
| Supportive Courses | ||
| Media and Management Skills Courses | ||
| Transferable Skill Courses |
| Week | Subjects | Related Preparation |
| 1 | Scalar product, norm, orthogonality. | Introductory functional analysis with applications by E. Kreyszig, Wiley,1989 |
| 2 | Linear Integral equations, Special types of kernels. Square integrable functions and kernels. Singular and nonlinear integral equations | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 1 |
| 3 | Connection with differential equation theory: Linear differential equations, Green’s and influence functions | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 2 |
| 4 | Integral Transforms, Fredholm equation of the second kind, Volterra equation of the second kind. | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 3 |
| 5 | Fredholm equation of the first kind, Volterra equation of the first kind. Fox’s integral equation. | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 3 |
| 6 | Method of successive approximations: Neumann series, Iterates and the resolvent kernels | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 4 |
| 7 | Integral equations with singular kernels: Generalization to higher dimensions, Green’s functions in two and three dimensions, Dirichlet’s problem, | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 5 |
| 8 | Hilbert Kernel, Hilbert transforms, Singular integral equation of Hilbert type | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 5 |
| 9 | Hilbert space: Euclidean space, Hilbert space of sequences | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 6 |
| 10 | Function space, Abstract Hilbert space | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 6 |
| 11 | Linear operators in Hilbert space: Linear integral operators, | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 7 |
| 12 | Bounded linear operators and completely continuous operators | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 7 |
| 13 | The resolvent: Resolvent equation, uniqueness theorems | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 8 |
| 14 | Fredholm equation in abstract Hilbert space | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 8 |
| 15 | Fredholm theory: Degenerate kernels, approximation by degenerate kernels, Fredholm formulae for continuous kernels | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications: Chapter 9 |
| 16 | Review of the Semester |
| Course Notes/Textbooks | Integral Equatios by B.L. Moiseiwitsch 7/e, Dowe Publications |
| Suggested Readings/Materials | Introductory functional analysis with applications by E. Kreyszig, Wiley,1989. Volterra Integral and Differential Equations by T.A. Burton, 2005. |
| Semester Activities | Number | Weigthing |
| Participation | ||
| Laboratory / Application | ||
| Field Work | ||
| Quizzes / Studio Critiques | ||
| Portfolio | ||
| Homework / Assignments | ||
| Presentation / Jury |
1
|
30
|
| Project | ||
| Seminar / Workshop | ||
| Oral Exams | ||
| Midterm |
1
|
30
|
| Final Exam |
1
|
40
|
| Total |
| Weighting of Semester Activities on the Final Grade |
2
|
60
|
| Weighting of End-of-Semester Activities on the Final Grade |
1
|
40
|
| Total |
| Semester Activities | Number | Duration (Hours) | Workload |
|---|---|---|---|
| Theoretical Course Hours (Including exam week: 16 x total hours) |
16
|
3
|
48
|
| Laboratory / Application Hours (Including exam week: '.16.' x total hours) |
16
|
0
|
|
| Study Hours Out of Class |
16
|
5
|
80
|
| Field Work |
0
|
||
| Quizzes / Studio Critiques |
0
|
||
| Portfolio |
0
|
||
| Homework / Assignments |
0
|
||
| Presentation / Jury |
1
|
22
|
22
|
| Project |
0
|
||
| Seminar / Workshop |
0
|
||
| Oral Exam |
0
|
||
| Midterms |
1
|
35
|
35
|
| Final Exam |
1
|
40
|
40
|
| Total |
225
|
|
#
|
Program Competencies/Outcomes |
* Contribution Level
|
|||||
|
1
|
2
|
3
|
4
|
5
|
|||
| 1 |
To develop and deepen his/her knowledge on theories of mathematics and statistics and their applications in level of expertise, and to obtain unique definitions which bring innovations to the area, based on master level competencies, |
-
|
-
|
-
|
-
|
X
|
|
| 2 |
To have the ability of original, independent and critical thinking in Mathematics and Statistics and to be able to develop theoretical concepts, |
-
|
-
|
-
|
-
|
X
|
|
| 3 |
To have the ability of defining and verifying problems in Mathematics and Statistics, |
-
|
-
|
-
|
-
|
X
|
|
| 4 |
With an interdisciplinary approach, to be able to apply theoretical and applied methods of mathematics and statistics in analyzing and solving new problems and to be able to discover his/her own potentials with respect to the application, |
-
|
-
|
-
|
X
|
-
|
|
| 5 |
In nearly every fields that mathematics and statistics are used, to be able to execute, conclude and report a research, which requires expertise, independently, |
-
|
-
|
-
|
-
|
X
|
|
| 6 |
To be able to evaluate and renew his/her abilities and knowledge acquired in the field of Applied Mathematics and Statistics with critical approach, and to be able to analyze, synthesize and evaluate complex thoughts in a critical way, |
-
|
-
|
-
|
-
|
X
|
|
| 7 |
To be able to convey his/her analyses and methods in the field of Applied Mathematics and Statistics to the experts in a scientific way, |
-
|
-
|
-
|
X
|
-
|
|
| 8 |
To be able to use national and international academic resources (English) efficiently, to update his/her knowledge, to communicate with his/her native and foreign colleagues easily, to follow the literature periodically, to contribute scientific meetings held in his/her own field and other fields systematically as written, oral and visual. |
-
|
-
|
-
|
-
|
X
|
|
| 9 |
To be familiar with computer software commonly used in the fields of Applied Mathematics and Statistics and to be able to use at least two of them efficiently, |
-
|
-
|
-
|
-
|
X
|
|
| 10 |
To contribute the transformation process of his/her own society into an information society and the sustainability of this process by introducing scientific, technological, social and cultural advances in the fields of Applied Mathematics and Statistics, |
-
|
-
|
-
|
X
|
-
|
|
| 11 |
As having rich cultural background and social sensitivity with a global perspective, to be able to evaluate all processes efficiently, to be able to contribute the solutions of social, scientific, cultural and ethical problems and to support the development of these values, |
-
|
-
|
-
|
-
|
X
|
|
| 12 |
As being competent in abstract thinking, to be able to connect abstract events to concrete events and to transfer solutions, to analyze results with scientific methods by designing experiment and collecting data and to interpret them, |
-
|
-
|
-
|
-
|
X
|
|
| 13 |
To be able to produce strategies, policies and plans about systems and topics in which mathematics and statistics are used and to be able to interpret and develop results, |
-
|
-
|
-
|
X
|
-
|
|
| 14 |
To be able to evaluate, argue and analyze prominent persons, events and phenomena, which play an important role in the development and combination of the fields of Mathematics and Statistics, within the perspective of the development of other fields of science, |
-
|
-
|
-
|
X
|
-
|
|
| 15 |
In Applied Mathematics and Statistics, to be able to sustain scientific work as an individual or a group, to be effective in all phases of an independent work, to participate decision-making process and to make and execute necessary planning within an effective time schedule. |
-
|
-
|
-
|
-
|
X
|
|
*1 Lowest, 2 Low, 3 Average, 4 High, 5 Highest
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