This course discusses some of the analytical and numerical techniques used in solving engineering problems. Most of the topics in this course deal with problems in mathematical physics.

• Ordinary Differential Equations
• Fourier Analysis
• Partial Differential Equations
• Integral Equations
• Variational Techniques
• Finite Element Method
  • Solution of Laplace Equation
  • Example Code
• Numerical Solution of the Heat Equation
  • Talk in PDF Format

Each topic has analytical and numerical parts. Almost all assignments will involve computer programming. But this course is not only about solving differential equations. One main objective is to understand the physics behind the equations.

• Stability of various numerical ODE solution methods
• Kapitza's Inverted Pendulum
• Series RLC circuit: LPF, BPF, and HPF Frequency responses
• RLC 2nd order LPF circuit: Unit step response

Solutions of the 2-D Laplace equation:

• B.C.: 0,0,0,1
• B.C.: 0,0,I,1
• B.C.: I,0,0,1
• B.C.: 0,I,0,1
• B.C.: I,0,I,x/a

Solutions of the 1-D heat equation:

• Infinite rod: The Heat Kernel
• Finite rod with both ends at zero temperature: Initial temperature distribution is triangular.
• Finite rod with both ends at zero temperature: Initial temperature distribution is constant.
• Finite rod with both ends insulated: Initial temperature distribution is linear ramp.
• Finite rod with one end at zero temperature and the other end insulated: Initial temperature distribution is constant.
• Finite rod with one end at zero temperature and the other end insulated: Initial temperature distribution is triangular.

Solutions of the 2-D Poisson equation:

• Uniform load: All sides fixed
• Uniform load: One side free
• Uniform load: Two sides free

• E. Kreyszig

  Advanced Engineering Mathematics

  John Wiley & Sons, Seventh Edition, 1993

• George B. Arfken, Hans J. Weber, Frank E. Harris

  Mathematical Methods for Physicists

  Academic Press, Seventh Edition, 2012

• Earl A. Coddington

  An Introduction to Ordinary Differential Equations

  Dover Publications, 1990

• T. W. Koerner

  Fourier Analysis

  Cambridge University Press, 1988

• R. V. Churchill

  Fourier Series and Boundary Value Problems

  McGraw Hill, New York, 1941

• W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery

  Numerical Recipes in C/FORTRAN

  Prentice Hall of India, New Delhi, 1994