Math 330 Fall 2022 Final Project
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Math 330
Fall 2022
Final Project
Due: Thursday, December 15th, 11:59 p.m.
Show all work and write your answers neatly for full credit. Your Matlab code and project report must be uploaded onto Canvas.
1. Gaussian Quadrature
(a) Find the abscissae xi and the weights wi of the three-point Gauss-Hermite quadrature formula
\− e −x2 f(x)dx ≈ w1 f(x1 ) + w2 f(x2 ) + w3 f(x3 ).
Use the fact that the Hermite polynomials ϕj (x) are orthogonal in the corresponding inner product
(f,g) = \− e −x2 f(x)g(x) dx.
Hermite polynomials are given by
ϕ0 = 1
ϕ 1 (x) = 2x
ϕj+1(x) = 2xϕj (x) − 2j ϕj −1 (x), j = 1, 2, 3, . . .
You can use Matlab or Mathematica to assist with integration if you choose to integrate La- grange interpolating polynomials to compute the weights. You must show work for setting up the integrals.
(b) Use your results from (a) to evaluate both
\− dx and \− dx
to eight significant digits. Also, compute the absolute error to 8 significant digits. You can use mathematical software to obtain the exact value of the integrals.
2. The ODE system given by
4y1 y2
y = α − y1 −
y = βy1 ( 1 − ),
where α and β are parameters, represents a simplified approximation to a chemical reaction. There is a parameter value βc = 3α/5 − 25/α such that for β > βc solution trajectories decay in amplitude and spiral in phase space into a stable fixed point, whereas for β < βc trajectories oscillate without damping and are attracted to a stable limit cycle. (This is called a Hopf bifurcation.)
(a) Set α = 10 and use the classical RK4 (p. 495) with a fixed step size h = 0.01 to approximate the solution starting at y1 (0) = 0, y2 (0) = 2, for 0 ≤ t ≤ 20. Do this for the parameter values β = 2 and β = 4. For each case plot y1 and y2 vs. t and y2 vs. y1 . Use the graphing template provided to produce nice graphs. Describe your observations.
(b) Investigate the situation closer to the critical value βc = 3.5 as β → βc from the right and left. (You may have to increase the length of the integration interval b to get a better look.) Provide graphs to justify your answers.
(c) Extra credit: Linearize the system about its equilibrium point and theoretically confirm your results from (a) by computing eigenvalues. You can use a symbolic manipulator.
2022-12-17