Real Analysis and Partial Differential Equations

Pre-Requisite: None
Contact Hours and Credits: (3 -0- 0) 3


To expose the students to the basics of real analysis and partial differential equations required for their subsequent course work.

Topics Covered:

Real number system. Sets, relations and functions. Properties of real numbers. Numerical sequences. Cauchy sequences. Bolzano-Weierstrass and Heine-Borel properties.

Functions of real variables. Limits, continuity and differentiability. Taylor’s formula. Implicit and inverse function theorems. Extrema of functions.

Reimann integral. Mean value theorems. Differentiation under integral sign. Improper and multiple integrals. Change-of-variables formula.

Sequences and series of functions. Pointwise and uniform convergence. Power series and Taylor series.

Laplace and Helmholtz equations. Boundary and initial value problems. Solution by separation of variables and Eigen function expansion.

Course Outcomes:

Students are able to

  • CO1: Develops an understanding for the construction of proofs and an appreciation for deductive logic.
  • CO2: Explore the already familiar properties of the derivative and the Riemann Integral, set on a more rigorous and formal footing which is central to avoiding inconsistencies in engineering applications.
  • CO3:Explore new theoretical dimensions of uniform convergence, completeness and important consequences as interchange of limit operations.
  • CO4: Develop an intuition for analyzing sets of higher dimension (mostly of the Rn type) space.
  • CO5: Solve the most common PDEs, recurrent in engineering using standard techniques and understanding of an appreciation for the need of numerical techniques.

Text Books:

  • Guenther, R.B. & Lee, J.W., Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice Hall, 1996.
  • Mattuck, A., Introduction to Analysis, Prentice-Hall, 1998.

Reference Books:

  • Kreyszig, E., Advanced Engineering Mathematics, John Wiley, 1999.
  • W. R. Parzynski & P. W. Zipse, Introduction to Mathematical Analysis, McGraw-Hill, (1/e), 1987.
  • G.B. Gustafson & C.H. Wilcox, Advanced Engineering Mathematics, Springer Verlag, 1998.