BCAM and UPV/EHU courses 2011-2012: Advanced aspects in applied mathematics
Topics on numerics for wave propagation
LAB1: HARMONIC OSCILLATORS
AURORA MARICA AND ENRIQUE ZUAZUA
Problem 1. Consider the harmonic oscillator equation:
(0.1)
x00 (t) + ω 2 x(t) = 0,
t ∈ [0, T ], x(0) = x0 , x0 (0) = x1 .
Set T = 4ωπ, ω = 1 and the following four initial data:
(0.2)
I) x0 = 1, x1 = 0;
II) x0 = 0, x1 = 1;
III) x0 = 1, x1 = 1;
IV ) x0 = 1, x1 = −1.
Task 1. Compute the explicit solution and plot it using Matlab.
Consider a general system of the form
(0.3)
x0 (t) = f (x(t), v(t)),
v 0 (t) = g(x(t), v(t)), t ∈ (0, T ),
x(0) = x0 , v(0) = v 0 .
To solve system (0.3) with Matlab we can proceed as follows:
Step 1. Introduce the following function and save it as funct.m
function der=funct(t,X)
der=[f(X(1),X(2));g(X(1),X(2))];
Step 2. The following Matlab program solves problem (0.3) using the ode45 function:
X0 = [x0 , v 0 ]; % vector of initial data
T = 4π; dt = T /1000;% the time step
divt = [0 : dt : T ];
[t0, sol] = ode45(@f unct, divt, x0);
Task 2. Solve problem (0.1) by using the above program. Plot the true solution x(t) and the numerical
one sol(:, 1) in the same figure. Plot sol(:, 1) against sol(:, 2) in the horizontal/vertical direction.
The following program draws the phase portrait (x, v) of system (0.3) for x ∈ [a, b] and v ∈ [c, d].
x = [a : hx : b]; v = [c : hv : d]; % the division of the intervals [a, b] and [c, d];
[X, V ] = meshgrid(x, v);
Xdot = f (X, V );
V dot = g(X, V );
quiver(X, V, Xdot, V dot)
Task 3: Draw the phase portrait of problem (0.1) for a = b = c = d = 2 and hx = hv = 0.2.
Problem 2. Consider the pendulum equation:
(0.4)
x00 (t) + ω 2 sin(x(t)) = 0,
t ∈ [0, T ], x(0) = x0 , x0 (0) = x1 .
Set T = 4π, ω = 1 and the following four initial data:
(0.5)
I) x0 = π/10, x1 = 0;
II) x0 = π/2, x1 = 0;
III) x0 = π, x1 = 0;
IV ) x0 = 3π/2, x1 = 0.
Repeat Tasks 2 and 3 in the previous problem.
Problem 3. Consider the damped harmonic oscillator:
(0.6)
x00 (t) + 2βx0 (t) + x(t) = 0,
t ∈ [0, T ], x(0) = x0 , x0 (0) = x1 .
Set T = 4π, x0 = 1, x1 = 0 and the following four values of β:
(0.7)
I) β = 0.05;
II) β = 0.5;
III) β = 1;
Repeat Tasks 1,2,3 in Problem 1.
1
IV ) β = 1.5.
2
AURORA MARICA AND ENRIQUE ZUAZUA
Problem 4. Consider the damped harmonic oscillator with driven force:
(0.8)
x00 (t) + 2βx0 (t) + x(t) = A cos(ωt),
t ∈ [0, T ], x(0) = x0 , x0 (0) = x1 .
Set T = 1/β, x0 = 1, x1 = 0, A = 1 and the following four values of β:
(0.9)
I) β = 1;
II) β = 1/10;
III) β = 1/50;
IV ) β = 1/200.
Repeat Task 2 of Problem 1 for all the following
six values ofpω: ω = 1/2, ω = 1/4, ω = (1 + ωd )/2,
p
ω = ωd , ω = (ωd + ωr )/2 and ωr , where ωd := 1 − β 2 and ωr := 1 − 2β 2 are the damping and resonance
frequencies.