COURSE UNIT TITLE

: THEORY OF HYPERBOLIC EQUATIONS AND SYSTEMS

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6033 THEORY OF HYPERBOLIC EQUATIONS AND SYSTEMS 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 the theory for the hyperbolic equations and systems. Weak solutions, energy estimates for the system of the first order hyperbolic equations are described. Existence and uniqueness theorems of the Cauchy problems, and initial boundary value problems for the hyperbolic equations and systems are studied.

Learning Outcomes of the Course Unit

1   Ability to understand the modern theory of the hyperbolic equations and systems.
2   Ability to express the fundamental concepts of the hyperbolic equations and systems.
3   Ability to use the main methods for solving the initial value problems for the hyperbolic equations and systems.
4   Ability to compute the fundamental solutions of the hyperbolic equations and systems.
5   Ability to applied solutions of the hyperbolic equations and systems 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 Weak derivatives, Sobolevs spaces, properties; Approximation, global approximation by smooth functions; Existence; Traces; Sobolevs inequalities; Poincares inequalities; Difference quotients.
2 Differentiability a.e. Fourier transform and alternate characterization of Sobolevs spaces. Other spaces of functions: Dual spaces, spaces involving time, mappings into better spaces.
3 Classical and weak solutions of hyperbolic equations. Energy estimate of the solution. Finite propagation speed. Energy Inequality. The domain of the dependence. Domain of influence.
4 Second order hyperbolic equations. Weak solutions of the initial / boundary value problem. Existence of weak solutions: The Hille-Yosida theorem. Smoothness of the solution.
5 Energy estimates for the second order hyperbolic equations and uniqueness theorem. Regularity. Energy conservation.
6 Hyperbolic systems of first-order equations. Weak solutions.
7 Midterm
8 Existence and uniqueness of the Cauchy problems, and initial boundary value problems for the hyperbolic systems.
9 Hyperbolic systems with constant coefficients.
10 Existence and uniqueness of the Cauchy problem for the Lames system of anisotropic elasticity.
11 Symmetric hyperbolic systems. Energy estimate for symmetric hyperbolic systems. Existence theorems of initial value problems for symmetric hyperbolic systems.
12 Uniqueness and stability estimate theorems .
13 Existence and uniqueness of the Cauchy problem for the Maxwells system of anisotropic electrodynamics. Methods of the construction of the solutions.
14 The construction of the asymptotic solutions for the hyperbolic equations.

Recomended or Required Reading

1.EVANS L. C., Partial Differential Equations, Graduate Studies in Mathematics, vol 19, American Mathematical Society, Institute for Advance Study, Providence, Rhode Island, 1998.
2. Ikawa M., Hyperbolic Partial Differential Equations and Wave Phenomena, vol. 189, American Mathematical Society, Providence,Rhode Island, 2000.
3.Caffarelli Luis A., Weinan E., Hyperbolic Equations and Frequency Interactions, vol.5, American Mathematical Society, Institute for Advance Study, Providence, Rhode Island, 1999.

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 8 80
Midterm 1 3 3
Final 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.1555555
LO.2555555
LO.355555555
LO.45555555
LO.5555555555

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

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