Differential Equations and Simulation

Basic information

Workload: 

45 hours

Syllabus: 

Introduction to floating point arithmetic. Numerical stability. Iterative methods for SEL: convergence analysis, stopping criteria. Conditioning. Solution of non-linear equations: Fixed point methods: convergence conditions, stopping criteria. Polynomial interpolation: Lagrange, Newton, Chebyshev polynomials, interpolation error. Curves adjustment. Numerical Integration: Newton-Cotes formulas, composite methods, Romberg method, Gaussian method. Differential equations: Mathematical models. The exponential matrix and solution of linear systems. Nonlinear ODE systems. Stability. Numerical integration of ODE: Taylor, Runge-Kutta, predictor-corrector, exponentials; convergence, A-stability, implementation. Computational methods for PDE. Stochastic simulation. Numerical integration of stochastic equations (EDEs): Euler, Milstein, Ito-Taylor. Types of approximation and convergence. Monte Carlo methods for EDEs. Computer simulation of EDEs. Applications.

Bibliography

Mandatory: 

•    Conte, S.D., de Boor, C. (1980). Elementary Numerical Analysis, an Algorithmic
•    Approach. McGraw-Hill.
•    Faire, D., & Burden, R. L. (2002). Numerical Methods. Brooks Cole.
•    Griffiths D., & Higham, D. (2010). Numerical Methods for Ordinary Differential Equations. Springer.
•    Higham D. (2004). An Introduction to Financial Option Valuation- Mathematics, Stochastics and Computation. Cambridge.
•    Stoer, J. & Bulirsch, R. (2002). Introduction to Numerical Analysis. Springer.
•    Sauer, T. (2011). Numerical Analysis. Pearson.