Course Identification

Complex analysis and partial differential equations
20182071

Lecturers and Teaching Assistants

Dr. Niv Sarig
Dr. Snir Ben Ovadia

Course Schedule and Location

2018
First Semester
Wednesday, 09:15 - 11:00, WSoS, Rm C
Wednesday, 11:00 - 12:00, WSoS, Rm A

Tutorials
Monday, 13:05 - 14:00, WSoS, Rm A
01/11/2017

Field of Study, Course Type and Credit Points

Chemical Sciences: Lecture; Elective; Core; 3.00 points

Comments

Has to be taken by all students of QM1
Courses location had change to:
9:15-11:00- FGS room C
11:00-12:00- FGS room A

Prerequisites

Mathematics for Chemists I or Basic Topics.

Basic knowledge in Linear Algebra, basic calculus (real differentiation and integration) and solvable Ordinary Differential Equations (ODE).
Possible courses which cover these prerequisites could be:

  1. For ODE, Math for Chemists 1 is preferable.
  2. Basic topics 1 is good as well but better for Linear algebra.
  3. Any undergraduate course on calculus and Linear algebra.

Restrictions

50

Language of Instruction

English

Attendance and participation

Expected and Recommended

Grade Type

Numerical (out of 100)

Grade Breakdown (in %)

50%
50%
Take home exam

Evaluation Type

Take-home exam

Scheduled date 1

N/A
N/A
-
N/A

Estimated Weekly Independent Workload (in hours)

4

Syllabus

List Of Topics:
 
(0) Crash overview and short technical introduction for Fourier series, Fourier transform and Delta function.
 
(1) Complex Analysis
  • Analytic functions.
  • Meromorphic function and residue theory.
  • Complex integration.
(2) Inner product spaces and self adjoint operators
 
(3) Sturm liouville theory
  • Partial differential equations - definition.
  • Generalized fourier series.
  • Separation of variables in bounded domains.
  • Solution of regular, periodic and singular PDE’s in bounded domains.
(4) Fourier transform and solution of PDE’s in unbounded domains.
 
(5) The Delta function - definition, convolution, linear time invariant operators and PDE’s with distributions.
 
(6) Calculus of variations
  • From Lagrangian and Hamiltonian to the Euler Lagrange equations
  • From calculus of variations to Strum Liouville theory.

Learning Outcomes

Upon successful completion of this course students should be able to:

  1. Solve partial differential equations in bounded and unbounded domain of Sturm Liouville type. 
  2. Understand the basic properties of complex analytic functions.
  3. Calculate complex integrals of several types of real and complex functions.
  4. Use the abstract notion of Inner product spaces in infinite dimensional vector spaces.
  5. Use the abstract notion of Hermitian operators and their relations to PDE.
  6. Understand the notion of generalized Fourier series and Fourier transform.
  7. Use the abstract notion of distributions - Delta functions, convolution and LTI operators.
  8. Solve boundary value PDE problems on bounded and unbounded domains with and without distributions.
  9. [Understand the notion of calculus of variation.
  10. Transform a variational problem to the corresponding Euler Lagrange equation.

Reading List

  1. F. John, Partial differential equations, New York : Springer, 1982, 4th edition. 
  2. [G.B. Arfken, H.J. Weber, Mathematical methods for physicists, Harcourt Academic Press, 2001, 5th edition. 
  3. W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley and Sons Inc, 2001, 7th edition. 
  4. B.M Marchesi, G. Pixton, D. Sabalka, A first course in complex analysis available on: http://math.sfsu.edu/beck/papers/complex.pdf 
  5. U.H Gerlach, Linear Mathematics in Infinite Dimensions - Signals, Boundary Value Problems and Special Functions available on: 
    http://www.math.osu.edu/ gerlach/math/BVtypset/BVtypset.html 
  6. Y. Pinchover , J. Rubinstein, Introduction to Partial differential equations, Cambridge university press (2005).

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