Tutorial Week 11
Worksheet Questions
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1. Solve the following 1D heat equation defined on the interval $0 \leq x \leq 2$, and having the initial condition $u(x,0) = x$ for $0\leq x \leq 2$. \begin{equation} \left\{ \begin{array}{lr} u_{t} = 4u_{xx}\\ u(0,t) = 0\\ u(2,t) = 0 \end{array} \right. \end{equation} 2. Solve the following 1D heat equation defined on the interval $0 \leq x \leq 2$, and having the initial condition $u(x,0) = x+1$ for $0\leq x \leq 2$. \begin{equation} \left\{ \begin{array}{lr} u_{t} = 4u_{xx}\\ u(0,t) = 1\\ u(2,t) = 5 \end{array} \right. \end{equation} 3. What family of trig function should you use in order to solve the following 1D heat equation defined on the interval $0 \leq x \leq 2$, and having the initial condition $u(x,0) = x$ for $0\leq x \leq 2$? Specify the function, $\sin(\omega x)$ or $\cos(\omega x)$, and the spatial frequencies $\omega$ as a function of an integer $n$. \begin{equation} \left\{ \begin{array}{lr} u_{t} = 4u_{xx}\\ u(0,t) = 0\\ u_{x}(2,t) = 0 \end{array} \right. \end{equation}
