Mathematical Methods for Engineers II

This graduate-level course is a continuation of Mathematical Methods for Engineers I (18.085). Topics include numerical methods; initial-value problems; network flows; and optimization.

• Lecture 1: Difference Methods for Ordinary Differential Equations
• Lecture 2: Finite Differences, Accuracy, Stability, Convergence
• Lecture 3: The One-way Wave Equation and CFL / von Neumann Stability
• Lecture 4: Comparison of Methods for the Wave Equation
• Lecture 5: Second-order Wave Equation (including leapfrog)
• Lecture 6: Wave Profiles, Heat Equation / point source
• Lecture 7: Finite Differences for the Heat Equation
• Lecture 8: Convection-Diffusion / Conservation Laws
• Lecture 9: Conservation Laws / Analysis / Shocks
• Lecture 10: Shocks and Fans from Point Source
• Lecture 11: Level Set Method
• Lecture 12: Matrices in Difference Equations (1D, 2D, 3D)
• Lecture 13: Elimination with Reordering: Sparse Matrices
• Lecture 14: Financial Mathematics / Black-Scholes Equation
• Lecture 15: Iterative Methods and Preconditioners
• Lecture 16: General Methods for Sparse Systems
• Lecture 17: Multigrid Methods
• Lecture 18: Krylov Methods / Multigrid Continued
• Lecture 19: Conjugate Gradient Method
• Lecture 20: Fast Poisson Solver
• Lecture 21: Optimization with constraints
• Lecture 22: Weighted Least Squares
• Lecture 23: Calculus of Variations / Weak Form
• Lecture 24: Error Estimates / Projections
• Lecture 25: Saddle Points / Inf-sup condition
• Lecture 26: Two Squares / Equality Constraint Bu = d
• Lecture 27: Regularization by Penalty Term
• Lecture 28: Linear Programming and Duality
• Lecture 29: Duality Puzzle / Inverse Problem / Integral Equations