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Syllabus ( MATH 675 )


   Basic information
Course title: Spectral Theory And Equations Of Mathematical Physics
Course code: MATH 675
Lecturer: Prof. Dr. Mansur İSGENDEROĞLU (İSMAİLOV)
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: Turkish
Mode of delivery: Face to face
Pre- and co-requisites: None
Professional practice: No
Purpose of the course: Demonstration the role of Spectral Theory to integrate of linear and non-linear equations of Mathematical Physics will be studied.
   Learning outcomes Up

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

  1. Gain the knowledge of spectral properties of diffrential operators

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    3. Design and conduct research projects independently

    Method of assessment

    1. Written exam
  2. Obtain the methods of integration for linear and non-linear equations of mathematical physics

    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

    Method of assessment

    1. Written exam
    2. Oral exam
    3. Homework assignment
  3. Explain the role of spectral theory for differential operators in mathematical physics

    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. Design and conduct research projects independently

    Method of assessment

    1. Written exam
    2. Oral exam
    3. Homework assignment
   Contents Up
Week 1: Linear Differential Expressions and Boundary Conditions, Differential Operator, Lagranges Formula, Adjoint Differential Expressions and Adjoint Boundary Conditions, Adjoint Differential Operator
Week 2: Eigenvalues, Eigenfonctions and Associated Functions of Differential Operator
Week 3: Green Functians and Inversion of a Differential Operator
Week 4: Boundary-value Problems involving a Parameter and their reduction to an Integral Equation
Week 5: Normalizing of a Boundary Conditions, Regular Boundary Conditions
Week 6: Asymptotic Behaviour of Eigenvalues and Eigenfonctions
Week 7: Expansion in term of Eigenfonctions of a Self-Adjoint Differential Operator, Expansion in term of Eigenfonctions and Associated Functions of a Differential Operator with Regular Boundary Conditions
Week 8: Midterm exam. The Solutions of Boundary-value Problems for Partial Differential Equations by Fouriers Method
Week 9: Integrable Nonlinear Evolition Equations with (1+1) and (2+1) dimensions, Solitary Waves and Solitons
Week 10: Lax Method for Characterization of Exact Solvability, Lax pair for Korteweg-de Vries (KdV)equation
Week 11: AKNS Method for Characterization of Exact Solvability, Lax pair for Sine-Gordon and Nonlinear Schrodinger Equation
Week 12: Scattering and Inverse Scattering Problems for One-Dimensional Schrodinger Equation on the line
Week 13: Scattering Data, Gelfand-Levitan-Marchenko Equation, Evolution of Scattering Data
Week 14: Scheme of Inverse Scattering Method, N-Soliton Solution of KdV Equation
Week 15*: ---
Week 16*: Final exam
Textbooks and materials:
Recommended readings: N. A. Naimark, Linear Differential Operators;
E. A. Coddington and N. Levinson, Theory of differential equations;
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform;
V. A. Marchenko, Sturm-Liouville operators and applications
  * 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: 8 40
Other in-term studies: 0
Project: 0
Homework: 3, 5, 7, 9, 11, 13 10
Quiz: 0
Final exam: 16 50
  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: 4 14
Practice, Recitation: 0 0
Homework: 8 6
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 20 1
Mid-term: 3 1
Personal studies for final exam: 20 1
Final exam: 3 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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