COURSE UNIT TITLE

: MATHEMATICAL THEORY OF INVERSE PROBLEMS

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6038 MATHEMATICAL THEORY OF INVERSE PROBLEMS ELECTIVE 3 0 0 8

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 elements of the theory of the integral geometry and tomography problems, inverse and ill-posed problems for hyperbolic, elliptic and parabolic equations.

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Learning Outcomes of the Course Unit

1   Ability to understand the modern theory of the integral geometry and tomography problems, inverse and ill-posed problems for partial differential equations.
2   Ability to express the fundamental concepts of the integral geometry and tomography problems, inverse and ill-posed problems for partial differential equations.
3   Ability to use the main methods for solving the integral geometry and tomography problems, inverse and ill-posed problems for partial differential equations.
4   Ability to compute solutions for the tomography, inverse and ill-posed problems.
5   Ability to use tomography, inverse and ill-posed problems for applied mathematical modeling.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Inverse problems concept: examples of formulating inverse problems. Well-posed and ill-posed problems.
2 Inverse Problem for Ordinary Differential Equations. Problems of determination of the right side and coefficients of linear ordinary differential equations and systems by their solutions. Inverse Sturm Liouville problem.
3 Inverse Problem for Hyperbolic Equations.
4 Inverse problem for the Klein-Gordon-Fock equation. Reducing the problem to integral equations.
5 Necessary and sufficient conditions for the inverse problem solvability. Relationship with Sturm-Liouville problem.
6 One-dimensional inverse problems for second-order hyperbolic equations. Problem of determining the coefficients in second-order hyperbolic equations.
7 Midterm
8 Inverse problem for the acoustic equation.
9 Inverse problem for parabolic and elliptical equations Inverse problem for the heat conductivity equation. Problem of determining the density of heat sources.
10 Problem of determining diffusion coefficiens. Relations among inverse problems for parabolic, elliptical, and hyperbolic type equations
11 Integral Geometry Problems . Determining the function of one variable from the integral of this function. The moment problem.
12 Inversion of the Radon transformation. The problem of computer tomography and its applications.
13 Maxwell system of electrodynamics and inverse problems for Maxwell system.
14 Lame system of isotropic elasticity and inverse problems for the Lame system.

Recomended or Required Reading

1. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Verlag, New York, 1996.
2. M.M. Lavrentev, K.G. Reznitskay, V.G. Yakhno, One-Dimensional Inverse Problems of Mathematical Physics, American Mathematical Society, Providence Rhode Island, 1986.
3. A.M. Denisov, Elements of the Theory of Inverse Problems, VSP, Utrecht, The Netherlands, 1999.
4. V.G. Romanov, Inverse Problems of Mathematical Physics, VSP, Utrecht, The Netherlands, 1987.

Planned Learning Activities and Teaching Methods

Lectures
Homeworks
Examinations
Assigments

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 dtermined 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 8 80
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 204

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.15555555
LO.255555555
LO.355555555
LO.455555555
LO.55555555

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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