Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 5

Divide $[a,b]$ into $N$ equal subintervals of width $h=(b-a)/N$, with $x_j=a+jh$. Linear interpolation between adjacent values of $f$ gives a trapezoid of signed area $h(f(x_{j-1})+f(x_j))/2$. Adding these areas produces the composite trapezoidal ruleFor continuous $f$, this tends to the integral as $h\to 0$; if $f$ has a bounded continuous second derivative, the error is at most $(b-a)h^2\max |f^{\prime \prime }|/12$.

The original PDF prints $n\to \infty$ while using $n$ as the bound summation index and $N$ as the upper limit. Literally this does not specify a varying upper limit. Interpreting the intended limit as $N\to \infty$, and relabelling the summation index $j$, gives the right Riemann sumFor $f(x)=x^4$, $a=0$, and $b=1$, the only difference from the composite trapezoidal rule is the endpoint weighting: $S_N=T_N+1/(2N)$. Hence

The given square wave agrees almost everywhere with an odd function on $(-\pi ,\pi )$; its assigned value at the single point $0$ does not affect any Fourier coefficient. Thus $a_0=a_n=0$, whileThe Fourier series is thereforeThe Dirichlet convergence theorem for Fourier series applies because the periodic extension is piecewise smooth. It gives $f(x)$ at each continuity point and the midpoint $0$ at the jumps, including $x=0$; it does not reproduce the assigned value $f(0)=-1$ there.

At the continuity point $x=\pi /2$, $\sin ((2r+1)\pi /2)=(-1{)}^r$ and $f(\pi /2)=1$. ThusNo evaluation at a discontinuity or rearrangement of the conditionally convergent series is involved.

A Fourier sine series automatically meets the zero-temperature boundary conditions because $\sin 0=\sin (n\pi )=0$. Equivalently, extend ${\Theta }_0$ oddly from $(0,L)$ to $(-L,L)$ and then periodically with period $2L$. Its Fourier series has only sine terms, withThe coefficient formula follows from the orthogonality identityFor piecewise smooth initial data the series equals ${\Theta }_0$ at interior continuity points and the average of its one-sided values at interior jumps. More general square-integrable data have convergence in L2 norm convergence. A classical solution continuous at the initial corners additionally requires ${\Theta }_0(0)={\Theta }_0(L)=0$, in agreement with the prescribed boundary conditions.

Substitute the proposed Fourier sine series into the heat equation:Projecting onto each sine using orthogonality gives $T_n^{\prime }=-D{k}_n^2{T}_n$ for every mode with $b_n\ne 0$. The initial condition fixes $T_n(0)=1$, since $b_n$ already describes the initial amplitude. HenceFor a mode with $b_n=0$, the value of $T_n$ is immaterial and the same convention is convenient. The positive thermal diffusivity $D$ damps every mode, with higher frequencies decaying faster. For square-integrable initial data the damping justifies termwise differentiation for $t>0$ and recovery of the initial data in L2 norm convergence as $t\downarrow 0$.

The left panel illustrates the Fourier series in part (b). The right panel is a self-generated example with $D=L=1$ and ${\Theta }_0(x)=\sin (\pi x)+\frac{1}{2}\sin (3\pi x)$, showing the two normal modes decaying at rates ${\pi }^2$ and $9{\pi }^2$.