这是一份southampton南安普敦大学MATH1009W1-01作业代写的成功案例
We examine a one-dimensional equation
$$
\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial^{2} u}{\partial x^{2}}+c^{2} u
$$
in the interval $0 \leq x \leq l$ with homogeneous initial conditions
$$
\left.u\right|{t=0}=\left.u{t}\right|{t=0}=0 $$ and boundary-value conditions $$ \left.u\right|{x=0}=\omega_{1}(t) ;\left.\quad u\right|{x=l}=\omega{2}(t) .
$$
It should be mentioned that the initial condition may always be reduced to homogeneous. The Bessel function of the imaginary variable $I_{0}\left(c \sqrt{t^{2}-x^{2}}\right)$ is the fundamental solution.
Placing the continuously acting sources, corresponding to this solution, at the ends of the interval $[0, l]$, we obtain, as may easily be seen, the ‘simple layer’ potentials:
$$
\begin{gathered}
\int_{0}^{t-x} \varphi(\tau) I_{0}\left(c \sqrt{(t-\tau)^{2}-x^{2}}\right) d \tau, \
\int_{0}^{t-(l-x)} \psi(\tau) I_{0}\left(c \sqrt{(t-\tau)^{2}-(l-x)^{2}}\right) d \tau,
\end{gathered}
$$
MATH1009W1-01 COURSE NOTES ：
We now determine functions $T_{n}(t)$, corresponding to the form of the wave $X_{n}(x)$. For this purpose, the value $\lambda_{n}$ is substituted into the equation for $T$ :
$$
T_{n}^{\prime \prime}+\frac{a^{2} \pi^{2}}{l^{2}} n^{2} T_{n}=0 .
$$
The general integral of this equation has the form::
$$
T_{n}(t)=B_{n} \sin \frac{\pi a n}{l} t+C_{n} \cos \frac{\pi a n}{l} t=A_{n} \sin \left(\frac{\pi a n}{l} t+\varphi_{n}\right) \text {, }
$$
where $B_{n}$ and $C_{n}$ or $A_{n}$ and $\varphi_{n}$ are arbitrary constants.
Using $X_{n}$ and $T_{n}$, we can write the final expression for all possible standing waves:
$$
u_{n}(x, t)=A_{n} \sin \left(\frac{\pi a n}{l} t+\varphi_{n}\right) \sin \frac{\pi n x}{l}
$$