19th Ave New York, NY 95822, USA

# 数学代写|An example of the heat equation 数值分析代考

## 数值分析代写

• Consider the heat equation
$$u_{t}=\kappa u_{x x}+f, \quad x \in(-1,1) \quad t \in(0,1)$$
• initial condition $u(x, 0)=0$
• boundary conditions $u(-1, t)=0$ and $u(1, t)=0$
• source term $f(x)=e^{x} \sin \left((5 x)^{2}\right)$
• $\kappa=1$
• Solving this system
$$0.15 \text { Solution to } u_{t}=u_{x x}+f, u(-1)=u(1)=0, \text { time }=0.0000$$
$$0.10$$
• Let us denote the steady-state solution $\bar{u}(x)$,
$$\bar{u}(x):=\lim _{t \rightarrow \infty}[u(x, t)]$$
supposing that such a limit exists. What equation does $\bar{u}(x)$ satisfy?
• Taking the limit $t \rightarrow \infty$ on the LHS of the heat equation, and assuming we can exchange the limit with the partial derivative
$$\lim {t \rightarrow \infty} \frac{\partial u}{\partial t}=\frac{\partial}{\partial t}\left[\lim {t \rightarrow \infty} u(x, t)\right]=\frac{\partial}{\partial t}[\bar{u}(x)]=0 \text {. }$$
• This is a simple ODE: $\bar{u}(x)$ has the general solution,
$$\bar{u}(x)=C_{1} x+C_{2},$$
and using the $B C s$, we obtain,
$$\bar{u}(x)=\frac{u_{r}-u_{\ell}}{L} x+u_{\ell} .$$
• Note that the initial condition $u(x, 0)=u_{0}(x)$ is nowhere to be seen: the initial condition was forgotten.

Matlab代写