\(\frac{1}{c^2}u_{tt} = u_{xx}\)
where \(u = u(x,t)\)
with\(u(0,t) = 0\)
\(u(l,t) = P\)
\(u(x,0) = 0\)
\(u_t(x,0) = 0\)
Solution Step 1. Transform to homogeneous conditionsSplit
\(u(x,t) = v(x) + w(x,t)\)
with\(v(0) = 0\)
\(v(l) = P\)
\(v_x(x) = \)
constantSee that
\(\frac{1}{c^2}w_{tt} = w_{xx}\)
with homogeneous boundry conditions\(w(0,t) = 0\)
\(w(l,t) = 0\)
\(w(x,0) = 0\)
\(w_t(x,0) = 0\)
Step 1 completed!Solution Step 2. See the trivial solution
w(x,t) = 0 is a solution to the wave equation
\(\frac{1}{c^2}w_{tt} = w_{xx}\)
(wave is at rest...)Step 2 accomplished!
Solution Step 3. Find the general solution of
\(w\)
[/b]Seperate
\(w(x,t) = X(x)T(t)\)
and solve!\(\frac{1}{c^2}w_{tt} = w_{xx} = \lambda\)
\(X(0) = 0\)
\(X(l) = 0\)
\(X'' - \lambda X = 0\)
\(T'' - c^2\lambda T = 0\)
Case 1: \(\lambda = 0\)
\(X(x) = a_1x + a_2\)
\(T(t) = b_1t + b_2\)
\(X(0) = 0\)
geeft \(a_2 = 0\)
\(X(l) = 0\)
geeft vervolgens \(a_1 = 0\)
dus \(X(x) = 0\)
en \(w(x,t) = 0\)
, wat de triviale oplossing was die we al hadden.Case 2:
\(\lambda > 0\)
\(X(x) = a_1e^{\sqrt{\lambda}x} + a_2e^{-\sqrt{\lambda}x}\)
\(T(t) = b_1e^{c\sqrt{\lambda}t} + b_2e^{-c\sqrt{\lambda}t}\)
\(X(0) = 0\)
geeft \(a_1 = a_2\)
\(X(l) = 0\)
geeft \(a_1 = 0\)
en \(a_2 = 0\)
dus \(X(x) = 0\)
en \(w(x,t) = 0\)
, wat wederom de triviale oplossing was die we al hadden.Case 3:
\(\lambda < 0\)
\(X(x) = a_1\sin{(\lambda}x)} + a_2\cos{(\lambda}x)}\)
\(T(t) = b_1\sin{(c\lambda t)} + b_2\cos{(c\lambda t)}\)
\(X(0) = 0\)
geeft \(a_2 = 0\)
\(X(l) = 0\)
geeft \(\lambda = \frac{\pi n}{\l}\)
dus \(X_n(x) = a_n\sin{(\frac{\pi nx}{\l})}\)
en \(T(t) = b_1\sin{(\frac{c \pi nt}{\l})} + b_2\cos{(\frac{c \pi nt}{\l})}\)
Algemene oplossingen voor \(X(x)\)
en \(T(t)\)
\(X(x) = \sum_{n=1}^{\infty} a_n\sin{(\frac{\pi nx}{\l})}\)
\(T(t) = \sum_{m=1}^{\infty}\left(b_m\sin{(\frac{c \pi mt}{\l})} + c_m\cos{(\frac{c \pi mt}{\l})}\right)\)
Step 3 successful!Solution Step 4. Terug naar
\(u(x,t)\)
[/b]\(u(x,t) = v(x) + w(x,t) = \frac{Px}{l} + X(x)T(t) = \frac{Px}{l} + \sum_{n=1}^{\infty} a_1\sin{(\frac{\pi nx}{\l})} \sum_{m=1}^{\infty}\left(b_m\sin{(\frac{c \pi mt}{\l})} + c_m\cos{(\frac{c \pi mt}{\l})}\right)\)
Mag je hier één som van maken?
\(u(x,t) = v(x) + w(x,t) = \frac{Px}{l} + X(x)T(t) = \frac{Px}{l} + \sum_{n=1}^{\infty} a_1\sin{(\frac{\pi nx}{\l})} \left(b_n\sin{(\frac{c \pi mt}{\l})} + c_n\cos{(\frac{c \pi mt}{\l})}\right)\)
De \(a_n\)
zou je dan weg kunnen laten, want die gaat dan op in \(b_n\)
en \(c_n\)
.\(u_t(x,0) = 0\)
geeft ...\(u(x,0) = 0\)
geeft ...Hoe gaat dit verder? Wie kan helpen?
Step 4 Not completed[/color]
