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Course Description:

The course consists of two main parts as follows:
  1. Partial Differential Equations (PDEs):
  2. In many cases we have two or more independent variables, so that the corresponding mathematical models involve partial, instead of ordinary, differential equations. In this part we discuss separation of variables as an important method for solving partial differential equations. Then we express the desired solution of the partial differential equation as a sum, usually an infinite series, formed from solutions of the ordinary differential equations. In many cases we ultimately need to deal with a series of sines and/or cosines, known as Fourier series. Then we show the use of separation of variables on a variety of problems arising from heat conduction, wave propagation, and potential theory. As a result of separating variables in a partial differential equation we consider a differential equation which is a sample of a large class of problems that are important in applied mathematics. These problems are known as Sturm-Liouville boundary value problems. Then we discuss the major properties and the solutions of these problems.
  3. Complex Variables and Applications:
  4. This part provides an introduction to complex analysis. Covered topics: Complex numbers, analytic functions, elementary functions, integrals, series, residues and poles, applications of residues, and mapping by elementary functions.

Text Books:

  1. Text book for PDE section: William E. Boyce , Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, Seventh Edition, John Wiley & Sons, Inc., 2001.
  2. Text book for Complex Variables section: James Ward Brown, and Ruel V. Churchill,, Complex Variables and Applications, Seventh Edition, McGrawHill.

References:

  1. Erwin Kreyszig, Advanced Engineering Mathematics , JOHN WILEY & SONS, INC., 2006.
  2. Richard A. Silverman, Complex Analysis with applications Dover Publications, Inc., New York, 1974.

Grading System: