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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5014 THEORY OF INTEGRAL EQUATIONS AND INTEGRAL TRANSFORMS ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

DOCTOR ALI SEVIMLICAN

Offered to

Mathematics (English) Mathematics (English)

Course Objective

This course gives an account of the multi-dimensional integral equations with the continuous and polar kernels as well as the Fourier, Radon and Hankel transformations and their properties.

Learning Outcomes of the Course Unit

1   Ability to understand the modern theory of multi-dimensional integral equations and their applications. 2   Ability to express the fundamental concepts of multi-dimensional integral equations techniques. 3   Ability to apply the main methods of solving the integral equations. 4   Ability to express the fundamental concepts of the Fourier, Radon and Hankel transformations. 5   Ability to apply the Fourier, Radon and Hankel transformations for solving partial differential equations.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Multi-dimensional Integral Equations with a continuos kernel. The Method of successive approximations. Existence and uniquness theorem for the Fredholm integral equation. Iterated kernals. Resolvents. 2 Solution of Fredholm integral equation in terms of resolvent. 1-D Volterra integral equation of the first and second kinds. Existence of a unique solution for the Volterra integral equation of the second kind; reduction of the Volterra integral equation of the first kind to the second kind. 3 Multi-dimensional integral equations with a polar kernel. Iterated kernals. Resolvents of the Fredholm integral equation, with a polar kernel. Existence and uniqueness theorem of the Fredholm integral equation with a polar kernel. 4 Fredholms theorems. Integral equations with a degenerate kernel. Fredholms theorems for the integral equations with a degenerate kernel. 5 Fredholms theorems for the integral equations with a continuos kernel. Consequences of Fredholm s theorems. 6 Fredholms theorems for the integral equations with a polar kernel. 7 Midterm 8 Multi-dimensional integral equations with an Hermitian continuous kernel. 9 The Hilbert-Schmit Theorem for an Continuous kernal. Extension of the Hilbert-Schmit theorem to integral equations with a Hermitian polar kernel. 10 The Fourier transform and its properties. 11 Fourier inversion theorem. The Fourier transform of the convolution . The range of the Fourier transform: Classical spaces, the Plancherel theorem, Riemann-Lebesgue lemma. 12 Analyticity of the Fourier transform: the Paley-Wiener theorem. 13 The Radon transform. The Radon transform . The inversion of the Radon transformation. 14 The Hankel transform. The definition of the Hankel transform. The connection with the Fourier transform.

Recomended or Required Reading

1. V.S. Vladimirov, Equations of Mathematical Physics, Marcel Dekker, INC., New York, 1971; 2. M.Reed, B.Simon, Fourier Analysis, Self-Adjointness, Academic Press, INC, New York, 1975; 3.A.D. Poularikas, The Transforms and Applications, CRC Press LLC, Boca Raton, Florida, 2000.

Planned Learning Activities and Teaching Methods

Lectures Homeworks Examinations Assignments

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 ASG ASSIGNMENT 2 MTE MIDTERM EXAM 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40

Further Notes About Assessment Methods

None

Assessment Criteria

Examinations

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

valery.yakhno@deu.edu.tr

Office Hours

Will be determined in the beginning of the term.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 3 39 Preparations before/after weekly lectures 13 3 39 Preparation for midterm exam 1 15 15 Preparation for final exam 1 25 25 Preparing assignments 10 5 50 Midterm 1 3 3 Final 1 3 3 TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 5 5 5 5 5 5 5 LO.2 5 5 5 5 5 5 5 5 LO.3 5 5 5 5 5 5 5 5 LO.4 5 5 5 5 5 5 LO.5 5 5 5 5 5 5 5 5 5 5