Heat Diffusion Equation - Using BCs to model as an orthonormal system

[physconomic]

Homework Statement

Consider heat equation
$\kappa \frac {\partial^2 \psi} {\partial x^2} = \frac{\partial \psi}{\partial t}$


$\kappa$ is positive, $x \subset [0,a]$, $\psi$ is real

For $t>0$: $\psi(t,0) = \psi_0$ at $x=0$

$\frac{\partial \psi}{\partial x}(t,a) = 0$ at $x=a$

$\psi(0,x) = 0$



We introduce $g_k(x) = \sqrt\frac{2}{a} sin(q_k x)$

where $q_k = \frac{\pi}{a}(k + \frac{1}{2})$

$k = 0, 1, ...$



b) Argue that the functions $g_k$ form an ortho-normal basis of the space $L_b ^2 ([0, a])$, of square integrable functions $f$ on $[0, a]$ with a Dirichlet boundary condition $f(0) = 0$ at $x = 0$ and a von Neumann condition $f'(a) = 0$ at $x = a$.



c) Based on the results in (a) (I've done this part - it's $\frac{d^2}{dx^2}g_k = -q_k^2g_k$) and (b) argue that the most general $\psi$ with the correct boundary conditions can be written as $\psi(t, x) = \psi_0+ \Sigma_0^\inf T_k(t)g_k(x)$. Find the solutions for the functions $T_k$.



d) Fix the remaining constants in your solution by imposing the initial condition. Compute the average value $\psi_{avg}(t)$ of $\psi(x, t)$ by averaging over x ∈ [0, a] and find an approximate equation for the time as a function of$r := (\psi_0 − \psi_{avg}(t))/\psi_0$.

Relevant Equations

Fourier series, Dirichlet, Von Neumann

I've tried to show b) by using the sine Fourier series on $[0,2a]$, to get $g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)$

Therefore $\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx$

These are equal therefore it is an orthonomal basis.

I'm not sure if this is correct so it would be great if somebody could help me by checking it and also letting me know how I could go about doing parts c and d.

Thank you

 

[vanhees71]

For (b) I'd rather argue by showing that $\mathrm{d}^2/\mathrm{d} x^2$ is a self-adjoint operator on the said Hilbert space. I'm also not sure what you mean by the sum over $n$.

 

[physconomic]

For (b) I'd rather argue by showing that $\mathrm{d}^2/\mathrm{d} x^2$ is a self-adjoint operator on the said Hilbert space. I'm also not sure what you mean by the sum over $n$.

Thanks for your reply. Can I ask how I would do this? I meant sum over k.