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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 6048 NUMERICAL FUNCTIONAL ANALYSIS 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 covers basic results of functional analysis and also some additional topics which are needed in theoretical numerical analysis. Applications of this functional analysis are given by considering, at length,numerical methods for solving differential equations and integral equations.

Learning Outcomes of the Course Unit

1   will be able to know basic definitions and theorems of functional analysis which are needed in theoretical numerical analysis. 2   will be able to know the analysis of iteration methods for nonlinear equations 3   will be able to know Sobolev spaces and weak formulations of boundary value problems 4   will be able to know the analysis numerical methods for solving integral equations of the second kind 5   will be able to know Ordinary differential equations in Banach spaces

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Linear spaces, Normed spaces Convergence, Banach spaces, Completion of normed spaces, Inner product spaces 2 Hilbert spaces, Orthogonality, Spaces of continuously differentiable functions, Holder spaces, Lp spaces ,Compact sets 3 Linear Operators on Normed Spaces, Continuous linear operators 4 The geometric series theorem and its variants 5 Some more results on linear operators, An extension theorem,Open mapping theorem, Principle of uniform boundedness, Convergence of numerical quadratures 6 Linear functionals, An extension theorem for linear functionals, The Riesz representation theorem 7 Midterm 8 Adjoint operators ,Types of convergence 9 Compact linear operators, Compact integral operators,Properties of compact operators,Integral operators on L2(a, b),The Fredholm alternative theorem 10 Nonlinear Equations and Their Solution by Iteration The Banach fixed-point theorem Applications to iterative methods Nonlinear equations 11 Linear systems Linear and nonlinear integral equations 12 Ordinary differential equations in Banach spaces 13 Newtons method Newtons method in a Banach space Applications, Conjugate gradient method for operator equations 14 Weak derivatives Sobolev spaces Properties

Recomended or Required Reading

Theoritical Numerical Analysis, Kendall Atkinson, Weimin han Numerical Functional Analysis, Colin W. Cryer

Planned Learning Activities and Teaching Methods

Lecture notes Presentation Homeworks

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE MTE * 0.20 + ASG * 0.40 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.20 + ASG * 0.40 + RST * 0.40

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

sennur.somali@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 midterm exam 1 20 20 Preparation for final exam 1 30 30 Preparing assignments 10 6 60 Midterm 1 3 3 Final 1 3 3 TOTAL WORKLOAD (hours) 194

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 4 5 5 5 3 5 4 LO.2 5 5 5 5 5 5 5 3 5 4 LO.3 5 5 5 5 5 5 5 3 5 4 LO.4 5 5 5 5 5 5 5 3 5 4 LO.5 5 5 5 5 5 5 5 3 5 4