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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5019 THEORY OF GENERALIZED FUNCTIONS AND APPLICATIONS 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

Offered to

Mathematics (English) Mathematics (English)

Course Objective

This course gives an account of the generalized functions theory. The fundamental solutions for the main partial differential operators are constructed by generalized functions.

Learning Outcomes of the Course Unit

1   Ability to understand the modern theory of the generalized functions and their applications. 2   Ability to express the fundamental concepts of the generalized functions techniques. 3   Ability to operate the generalized functions tools. 4   Ability to state correctly problems for the main partial differential equations. 5   Ability to construct the fundamental solutions for the main partial differential equations. 6   Ability to the generalized solutions for modeling real processes and phenomena.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Elements of the generalized functions theory. Space of test functions, examples. Space of generalized functions, examples. Regular and singular generalized functions, examples. 2 Dirac delta function. Du Buis Reymonds Lemma. 3 Operations with generalized functions. Composition of generalized and infinitely differentiable functions, examples. Multiplication of a generalized function by an infinitely differentiable function, examples. 4 Differentiation of generalized functions: derivatives of generalized functions, properties of generalized derivatives. Examples. 5 Definition of the direct product. Properties of the direct product: commutativity, continuity, associativity, differentiation, translation, multiplication. 6 Definition of convolution of generalized functions, the condition for the existence of the convolution. Properties of the direct product. 7 Midterm 8 Generalized functions of slow growth (Tempered distribution) . Space of test functions of slow growth, examples. Space of generalized functions of slow growth L.Schwartzs theorem. Examples of generalized functions of slow growth. 9 Structure of generalized functions with point support. 10 Fourier transform of test functions of slow growth. Examples. 11 Fourier transform of generalized functions of slow growth. Examples. 12 Properties of the Fourier transform . Differentiability of the Fourier transform, examples. Fourier transform of a translation, examples. Fourier transform of a direct product, examples. 13 Fourier transform of generalized functions with compact support Fourier transform of a convolution. Examples. 14 Fundamental solutions of ordinary and partial differential operators.

Recomended or Required Reading

1. Vladimirov V.S., Equations of Mathematical Physics, Pergamon Press, Oxford, 1963; 2. Zauderer E., Partial Differential Equations of Applied Mathematics, John Wiley & Sons, New York, 1989 ; 3. Courant R. Methods of Mathematical Physics, Vol.2, Wiley-Interscience, New York, 1962.

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 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

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

To be announced.

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 final exam 1 25 25 Preparation for midterm exam 1 15 15 Preparing assignments 10 5 50 Final 1 3 3 Midterm 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 LO.2 5 5 5 5 5 5 5 5 LO.3 5 5 5 5 5 5 5 5 5 5 LO.4 5 5 5 5 5 5 5 5 5 LO.5 5 5 5 5 5 5 5 LO.6 5 5 5 5 5 5 5 5 5