Math 405/607E (Numerical Methods for Differential Equations), Fall 2011

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• Class: Mon Wed Fri 13:00-13:50 in Math 203

• Instructor
  • Lisa Gordeliy
  • gordeliy(at)math(dot)ubc(dot)ca

• Office hours: in LSK 100D, Mon 11am-12noon and 3-5pm, Wed 11am-12noon, or by appointment via email.

• Course outline

Final Exam

• Final exam from past years

• Final exam description and grading scheme

Course notes

• MATLAB demo 1, MATLAB demo 2

• Interpolation and approximation notes, Numerical integration notes and Notes on numerical methods for ODEs, PDEs, and Wave equation notes by Prof. A. Peirce

• Course notes and homework solutions from previous years are available at Professor Anthony Peirce's website

Lectures

• Lecture 1

• Interpolation notes (A. Peirce): Lecture 2 (p. 5-6), Lecture 3 (p. 7), Lecture 4 (p. 9-11), Lecture 5 (p. 7-8, 13), Lecture 6 (p. 13-14), Lecture 7 (p. 14, 16-17) and Runge example, Lectures 8 - 11 (p. 18-28, and notes for spline construction)

• Numerical differentiation: Lecture 12

• Numerical integration notes (A. Peirce): Lectures 13 - 16

• Notes on numerical methods for ODEs (A. Peirce): Lecture 17 (p. 1-3), Lecture 18 (p. 4-5), Lecture 19 (p. 8-9, Forward Euler example for y' = -5y, y(0) = 1 with stability and instability), Lecture 20 ( Backward Euler and solution of nonlinear equations), Lecture 21 (p. 10-13), Lecture 22 (p. 13-16 and stiff ODE example), Lecture 23 (p. 18-21), Lecture 24 (overview of p. 21-31)

• Notes on numerical methods for PDEs (A. Peirce): Lecture 25 (p. 1-3), Lecture 26 (p. 4-6, 10), Lecture 27 (p. 10-11 and a MATLAB example of a BVP solved by finite differences using the Newton's method), Lectures 28 - 30 (tridiagonal systems (interpolation notes p. 29) and iterative methods for linear systems p. 15-21), Lectures 31 - 32 (solution of heat equation using finite differences and discussion of problem 2 of the Assignment)

• Wave equation notes (A. Peirce): Lectures 32 - 35 (and discretization of the wave equation using Crank-Nicolson in time / central difference in space)

• Lecture 36: finite element method (FEM)

Assignments

• Assignment 1 (0), due Sept 23 in class. MATLAB codes: ddifftable.m, difftable.m

• Assignment 2 (1), due Oct 12 in class. MATLAB codes: fddiff.m, DemoHermite.m. Expressions for Hermite functions and their derivatives, as well as linear systems for spline construction in problem 3, are given here. SOLUTIONS

• Assignment 3 (2), due Oct 26 in class. For problem 3, you can modify the following MATLAB codes: composite trapezoidal quadrature and testtrap.m - application of the composite trapezoidal quadrature with N subintervals to integrate f(x) = sin(5.*x) on [0,1]. Save these two codes to the same folder and run the code testtrap.m. You will see that by changing N, the quadratic convergence rate is obtained.
SOLUTIONS to problems 3 - 5

• Assignment 4 (3), due Nov 16 in class. euler.m, BackEuler.m, ODEsystem.m and a stiff ODE example SOLUTIONS

• Assignment 5 (4), due Nov 30 in class. FDnewton.m Equations to set up the solution in Problem 3(c). MATLAB code to plot the wave in Problem 3(c). SOLUTIONS

Projects

• Project I SOLUTIONS

• Project II, due Dec 2 in class. UPDATE: The analytic solution for the logistic equation is y(x) = 100/(1+19*exp(-200*x)).

• Project III for graduate students: due Dec 2 in class. A research project, discuss it with your advisor. This should be a problem that you want to solve using the numerical tools we learn in class, you may wish to use: interpolation, integration, finite differences, ODE numerical solutions, or PDE numerical solutions (to be covered in the second half of November in class).

Textbooks

• "Applied numerical analysis" by Curtis F. Gerald, Patrick O. Wheatley

• "Numerical analysis" by Richard L. Burden, J. Douglas Faires (Requested and expected to arrive at I.K. BARBER LEARNING CENTRE circulation reserve.)

MATLAB resources

• Lab instructions UPDATED September 13