# Applied Mathematics (MATH463) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Applied Mathematics MATH463 Area Elective 4 0 0 4 8
Pre-requisite Course(s)
Math 262 Ordinary Differential Equations
Course Language English Elective Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Discussion, Question and Answer, Problem Solving. The course is divided into two parts: Integral Equations and Calculus of Variations. In the first part, the course aims to present the basic 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-Schimidt theory, the Neumann series and Fredholm theory. The students who succeeded in this course; At the end of the course the students are expected to: 1- students are expected to know and understand various ideas, concepts and methods from applied mathematics and how these ideas may be used in, or are connected to, the fields of engineering and mathematics. 2 students will be able to apply various methods to solve a range of problems from applied mathematics and engineering - including: Integral equations, Green’s function and Calculus of Variations. Calculus of variations and applications, integral equations and applications.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 1. Week Maxima and minima of one variable and multivariable functions. The subject of calculus of variations. 2. Week The simplest case of variational problems. Necessary condition for the existence of an extremum: the Euler equation. Extremals. 3. Week Natural boundary conditions and transition conditions. Function spaces and functionals. 4. Week The concept of variation of functionals. A case of integrals depending on functions of two variables. 5. Week The more general case of variational problems. Variational problems with variable endpoints. 6. Week Application to Sturm-Liouville problems. Application to mechanics: Hamilton’s principle, Langrange’s equations, Hamilton’s canonical equations. 7. Week Basic Definitions.Fredholm and Volterra integral equations. 8. Week Midterm Exam 9. Week Relations between differential and integral equations. 10. Week The Green’s function. 11. Week Fredholm equations with separable kernels. 12. Week Hilbert-Schimidt theory. 13. Week Iterative methods for solving an integral equation of second kind. The Neumann series. 14. Week Fredholm theory.Singular integral equations. Special devices for solving some integral equations. 15. Week Methods for obtaining approximate solutions of integral equations. 16. Week Final Exam

### Sources

Course Book 1. F. B. Hildebrand, Methods of Applied Mathematics, 2nd Edition, 1965, Prentice – Hall, Englewood Cliffs. 2. 1] I. M. Gelfand and S. V. Fomin, Calculus of Variations, 1963, Prentice – Hall, Englewood Cliffs. [2] W. V. Lovitt, Linear Integral Equations, 1924, McGraw – Hill, New York.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments - -
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 60
Final Exam/Final Jury 1 40
Toplam 3 100
Percentage of Semester Work 100 100

### 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 Can apply gained knowledge and problem solving abilities in inter-disciplinary research
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
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
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility
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
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)
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach.
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 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 25 50
Prepration of Final Exams/Final Jury 1 35 35