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Syllabus ( MATH 548 )


   Basic information
Course title: Numerical Analysis of Partial Differential Equations
Course code: MATH 548
Lecturer: Prof. Dr. Mansur İSGENDEROĞLU (İSMAİLOV)
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 2,1, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: Numerical Analysis
Professional practice: No
Purpose of the course: In this course, the classical numerical methods for partial differential equations as well as convergence properties of these methods will be studied. The course will provide some fundamental knowledge and experience of working with numerical methods necessary for researchers in the field of applied mathematics and several related disciplines.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Develop and apply numerical solution methods to partial differential equations.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    6. Effectively express his/her research ideas and findings both orally and in writing

    Method of assessment

    1. Written exam
    2. Homework assignment
  2. Analyse the numerical tecqniques for convergence using analytical methods.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Acquire scientific knowledge and work independently
    3. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.
    5. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  3. Have knowledge on the discretisation, and development of efficient methods.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Design and conduct research projects independently
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Develop mathematical, communicative, problem-solving, brainstorming skills.
    8. Effectively express his/her research ideas and findings both orally and in writing
    9. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  4. Obtain approximate solutions to modeled life problems.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Work effectively in multi-disciplinary research teams
    6. Design and conduct research projects independently
    7. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    8. Develop mathematical, communicative, problem-solving, brainstorming skills.
    9. Effectively express his/her research ideas and findings both orally and in writing
    10. Defend research outcomes at seminars and conferences.
    11. Write progress reports clearly on the basis of published documents, thesis, etc
    12. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Term paper
   Contents Up
Week 1: Finite difference methods, analysis and properties of discrete operators
Week 2: Consistency, stability and convergence of the finite difference methods
Week 3: Finite difference methods on curved boundaries
Week 4: Distributional partial derivatives, Sobolev embedding theory
Week 5: Negative Sobolev spaces, Important inequalities
Week 6: Weak solutions, Linear elliptic PDE's
Week 7: Finite elements, Galerkin method, Method of Weighted residuals
Week 8: Basis functions, Interpolation errors
Week 9: Midterm exam
Week 10: L2 and negative norm estimates, Aubin-Nitsche Theorem
Week 11: Applications: Stokes problem, Linear elasticity problem
Week 12: Spectral methods, Trigonometric polynomials
Week 13: Fourier spectral method
Week 14: Spectral Galerkin method, Spectral Tau method
Week 15*: Spectral collocation method
Week 16*: Final exam
Textbooks and materials: S. H. Lui, Numerical Analysis of Partial Differential Equations, Wiley, 2011.
Recommended readings: K. W. Morton, D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 2005.

J. W. Thomas, Numerical Partial Differential Equations, Springer-Verlag, 1995.

M. K. Jain, Numerical Solution of Differential Equations, Wiley Eastern Limited, 1984.
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 9 30
Other in-term studies: 0
Project: 14 30
Homework: 2,4,6,8,10,12 10
Quiz: 0
Final exam: 16 30
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 0 0
Practice, Recitation: 1 14
Homework: 2 6
Term project: 5 14
Term project presentation: 2 1
Quiz: 0 0
Own study for mid-term exam: 10 2
Mid-term: 2 1
Personal studies for final exam: 10 2
Final exam: 2 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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