$($a$)$ For $u(x,t)$ defined on the domain of $0x1$ and $t0,$ solve the PDE

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^{2}u}{del{x}^2}$

with the boundary conditions,

$($i$)u_x(0,t)=0,$

$($ii$)u_x(1,t)=0,$

$($iii$)u(x,0)=P(x)$

where

$P(x)=\frac{[1-cos(2x){]}^8}{256}+\frac{1-cos(x)}{4}$

Plot the solution, $u(x,t),$ as a function of $x$ at $t=0,0\mathrm{.}003,0\mathrm{.}01,$ and $0\mathrm{.}1.$ Please collect all four

curves in a single plot.

$($b$)$ Repeat $($a$)$ but now solve the system with the $1$st boundary condition changed to

$($i$)u(0,t)=0.$

The $2$ nd and $3$ rd boundary conditions remain the same as in Part $($a$).$ Plot the solution, $u(x,t),$ as a

function of $x$ at $t=0,0\mathrm{.}003,0\mathrm{.}01,$ and $0\mathrm{.}1.$ Please collect all four curves in a single plot.

$($c$)$ Define heat flux $(x,t)$ as $-=-$del$\frac{u}{d}$elx. Based on your solution for Part $($b$),$ compute heat

flux at $x=0$ as a function of $t$ and make a plot of $(0,t)$ over the range of $0t0\mathrm{.}2.$ Please

use a sufficiently small increment of $t$ to ensure that the plot is smooth.

Note for Problem $1$:

$($i$)$ We expect the solution to be expressed as an infinite series. A truncation of the infinite series is

needed to numerically compute the series in order to plot the solution. It is your job to determine

the appropriate number of terms to keep. As a useful measure, the solution at $t=0$ should match

the given initial state in the $3$rd boundary condition. If they do not match, either the solution is

wrong or more terms need to be retained in the series. This remark applies to all future homework

problems that require the evaluation of an infinite series.

$($ii$)$ For the evaluation of the expansion coefficients in the infinite series, there is no need to carry

out the integrations analytically. $($If you wish to do so$,$ please feel free to use an online integrator

or similar software such as Wolfram Alpha. No need to do it by hand.$)$ It is perfectly acceptable

$($actually$,$ recommended for this problem$)$ to evaluate the integrals numerically using Matlab. See

Additional Note in the last page for an example of using Matlab to numerically evaluate an integral.