Solution Found!
Solve the boundary value problems in 1 through 12.
Chapter , Problem 1(choose chapter or problem)
Solve the boundary value problem
\(\begin{array}{l}
u_{t}=3 u_{x x}, 0<x<\pi, t>0 ; u(0, t)=u(\pi, t)=0, \\
u(x, 0)=4 \sin 2 x
\end{array}\)
Questions & Answers
QUESTION:
Solve the boundary value problem
\(\begin{array}{l}
u_{t}=3 u_{x x}, 0<x<\pi, t>0 ; u(0, t)=u(\pi, t)=0, \\
u(x, 0)=4 \sin 2 x
\end{array}\)
Step 1 of 5
We know that the boundary value problem
\(\begin{array}{l}
\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},(0<x<L, t>0) \\
u(0, t)=u(L, t)=0, u(x, 0)=f(x)
\end{array}\)
has the formal series solution
\(u(x, t)=\sum_{n=1}^{\infty} b_{n} \exp \left(-n^{2} \pi^{2} k t / L^{2}\right) \sin \frac{n \pi x}{L}\)
where
\(b_{n}=\frac{2}{L} \int_{0}^{L} f(x) \sin \frac{n \pi x}{L} d x\),
Fourier sine coefficient corresponding to the function f(x).