March Lectures

Week of March 1-5

Mar 1: trapezoidal rule and error formula; transforming quadrature rules to other intervals.

Mar 3: Simpson's rule and its degree of accuracy.

Tutorial: exercises of integration (midpoint, trapezoidal, Simpson); corrected trapezoidal rule and error formula.

Week of March 8-12

Mar 8: error in Simpson's rule; convergence of polynomial interpolatory quadrature rules; Newton-Cotes quadrature rules.

Mar 10: composite quadrature rules and error formulas; composite midpoint rule and error formula; other composite rules and error formulas.

Tutorial: error estimators for quadrature rules; examples of errors in quadrature rules; adaptive quadrature.

Week of March 15-19

Mar 15: Gauss quadrature rules.

Mar 17: infinite and semi-infinite integrals; singularities.

Tutorial: examples of Gauss quadrature rules.

Week of March 22-26

Mar 22: introduction: DEs, ODEs, PDEs and IVPs; existence and uniqueness of solution of an IVP for an ODE; second order ODEs and BVPs.

Mar 24: nth order ODEs and IVPs for ODEs; stability of ODEs -- Jacobian.

Tutorial: examples of ODEs, IVPs and Jacobian of a system of ODEs.

Week of March 29 - April 2

Mar 29: numerical methods for first order IVPs for ODEs; forward Euler's method: global and local errors; stability of the numerical method.

Mar 31: example of forward Euler's method; order of a numerical method for IVPs-ODEs; stiff ODEs.

Tutorial: no tutorial (Good Friday).

Week of April 5-9

Apr 5: backward Euler's method; implicit methods; trapezoidal method; stability and order.

Apr 7: relation between ODEs and quadrature rules. Exam preparation.

Tutorial: Runge-Kutta methods; modified Euler's method; examples of Runge-Kutta methods.