# Separation of Variables with functions which are not eigenfunctions

I'm studying a book named 'Computational Techniques for fluid dynamics' by Fletcher , and it solves a heat conduction problem with mixed boundary conditions.

\frac{\partial T}{\partial t}-\alpha \frac{\partial^2 T}{\partial^2 x}=0~, \qquad \begin{aligned} &\text{B.C.}\quad \left\{ \begin{aligned} \frac{\partial T}{\partial x}(0,t) &= -2 \\ T(1,t) &=1 \end{aligned} \right.\\ &\text{I.C.} \quad T(x,0) =3-2x-2x^2+2x^3 \end{aligned}

And it formulates the form of solution as

$$T=3-2x+b_0+\sum_{j=1}^J{a_j \sin(2\pi j x )+b_j \cos(2\pi j x)}$$

I cannot understand how it comes out. I can see how 3-2x comes from the steady state solution, but what about Fourier series? They doesn't even satisfy boundary conditions individually! However, the method works, I mean, if I plug in that form in the heat equation and let the computer solves for all coefficients a and b, I can see the numerical solution fits exactly on the exact solution.

So my question is, how non-eigenfunction can works as a component of separation of variables, and if it works, why the book had to separate steady state solution at all? Why it just didn't assume $$T=b_0+\sum_{j=1}^J{a_j \sin(2\pi j x )+b_j \cos(2\pi j x)}$$ and let computer do all the job? (I tried that, but it failed, and I cannot see what's the difference)

So my question is, how non-eigenfunction can works as a component of separation of variables, and if it works, why the book had to separate steady state solution at all? Why it just didn't assume $$T=b_0+\sum_{j=1}^J{a_j \sin(2\pi j x )+b_j \cos(2\pi j x)}$$ and let computer do all the job? (I tried that, but it failed, and I cannot see what's the difference)

Since we aren't requiring $$T(0,t) = T(1,t)$$, this isn't the general form for a bounded function on $$[0,1]$$. The general form is $$T=a_0x + b_0+\sum_{j=1}^J{a_j \sin(2\pi j x )+b_j \cos(2\pi j x)},$$

where $$a_0$$ is needed since the rest of the function is periodic. You'll note this is identical to the trial solution from the book except they chose $$a_0 = -2$$ in advance.

(That $$\partial T/\partial t$$ is supposed to be $$\partial T/\partial x$$, right?)

The function $$u(x,t) = T(x,t) - (3-2x)$$ (i.e., the deviation from the steady state) satisfies the heat equation with homogeneous boundary conditions $$u_x(0,t)=0$$, $$u(1,t)=0$$, so it makes sense to look for a Fourier series solution for $$u$$. Once you have that, the final solution will $$T(x,t)$$ will be the steady state $$3-2x$$ plus the Fourier series $$u(x,t)$$.

• You mean $u(1,t) = 0$, right? – eyeballfrog Apr 18 '19 at 18:26
• Oops! Yes, of course. Corrected now. Thank you. – Hans Lundmark Apr 18 '19 at 19:06
• Yeah B.C. should be derivative w.r.t. x. My question is, Fourier series solutions we are looking for should satisfy homogeneous boundary conditions, right? but sin(2pi j x) and cos(2 pi j x) only satisfy one of them each, and I cannot understand how they can be used as if they ar e eigenfunction. Standard eigenfunction should look like something like cos((2n+1)/2*pijx), I thought. – Septacle Apr 18 '19 at 20:29