 MATHEMATICS (TURKISH, PHD) PhD TR-NQF-HE: Level 8 QF-EHEA: Third Cycle EQF-LLL: Level 8

# Course Introduction and Application Information

 Course Code Course Name Semester Theoretical Practical Credit ECTS MAT6018 Numerical Solutions of Differential Equations II Spring 3 0 3 8
 This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

### Basic information

 Language of instruction: Turkish Type of course: Departmental Elective Course Level: Mode of Delivery: Face to face Course Coordinator : Assoc. Prof. ERSİN ÖZUĞURLU Recommended Optional Program Components: None Course Objectives: To see the applications and numerical solution of the partial differential equations.

### Learning Outcomes

 The students who have succeeded in this course; The students who succeeded in this course; o will be able to understand basic finite difference methods for partial differential equations. o will be able to solve numerically any given linear or nonlinear partial differential equation. o will be able to understand the concepts of consistency, stability, and convergence. o will be able to solve partial differential equations bu using using a computer program (C, C , Fortran, Matlab). o will be able to discuss the consistency, convergence and stability for schemes. o will be able to do error analysis.

### Course Content

 This course focuses on the fundamentals of modern and classical numerical techniques for linear and nonlinear partial differential equations, with application to a wide variety of problems in science, engineering and other fields. The course covers the basic theory of scheme consistency, convergence and stability and various numerical methods.

### Weekly Detailed Course Contents

 Week Subject Related Preparation 1) Hyperbolic Partial Differential Equations: Finite difference methods. 2) Consistency, stability, and convergence 3) The Lax Richtmyer equivalence theorem 4) The Courant Friedrichs Lewy condition. Von Neumann Analysis 5) Order of accuracy Multistep schemes. Dissipation and dispersion. 6) Parabolic Partial Differential Equations: Finite difference methods 7) Parabolic systems in higher dimensions 8) ADI methods 9) Elliptic Partial Differential Equations: Regularity and maximum principles 10) Finite difference methods, Linear iterative methods. 11) Multigrid methods 12) Rigorous convergence analysis 13) Error estimates 14) Matrix method for stability analysis, Spectral methods

### Sources

 Course Notes / Textbooks: Partial Differential Equations with Boundary Value Problems by Larry C. Andrews. Numerical Solution of Partial Differentail Equations by K.W. Morton and D.F. Mayers Numerical Solution of Partial Differential Equations: Finite Difference Methods by G.D. Smith References: Partial Differential Equations. Lawrence C. Evans Applied Partial Differential Equations Paul DuChateau, David Zachmann Applied Partial Differential Equations Richard Haberman Applied Partial Differential Equations John R. Ockendon, Sam Howison, John Ockendon, Andrew Lacey, Alexander Movchan

### Evaluation System

 Semester Requirements Number of Activities Level of Contribution Quizzes 5 % 15 Midterms 2 % 45 Final 1 % 40 Total % 100 PERCENTAGE OF SEMESTER WORK % 60 PERCENTAGE OF FINAL WORK % 40 Total % 100