# Applied Mathematics (MATH587) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Applied Mathematics MATH587 3 0 0 3 5
Pre-requisite Course(s)
Math 262 Ordinary Differential Equations
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Lecture, Discussion, Question and Answer, Problem Solving. 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. The students who succeeded in this course; 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. 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: 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. 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

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 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.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 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.
5 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.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 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.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
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 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