##\displaystyle \frac{\partial u(x,t)}{\partial t}-p(x) \frac{\partial^2 u(x)}{\partial x^2}-\frac{\partial p(x)}{\partial x}\frac{\partial u(x)}{\partial x}=f(x,t)##.(adsbygoogle = window.adsbygoogle || []).push({});

With periodic boundary conditions: ##u(x,t)=u(x+2\pi,t)##. The right hand side is also a periodic function of space, and it can be saparated: ##f(x,t)=f(x)T(t)##. I wanted to do this by means of Fourier series, so I've expanded everything in Fourier series:

##\displaystyle u(x)=\sum_{k=-\infty}^{\infty} \hat u_k e^{ikx},\frac{\partial u(x)}{\partial x}=\sum_{k=-\infty}^{\infty} ik \hat u_k e^{ikx},\frac{\partial^2 u(x)}{\partial x^2}=-\sum_{k=-\infty}^{\infty} k^2 \hat u_k e^{ikx},f(x)=\sum_{k=-\infty}^{\infty} \hat f_k e^{ikx},p(x)=\sum_{k=-\infty}^{\infty} \hat p_k e^{ikx},\frac{\partial p(x)}{\partial x}=\sum_{k=-\infty}^{\infty} ik \hat p_k e^{ikx}##.

So when I replace all this in the differential equation, with the aim to get a differential equation for the ##\hat u_k## in Fourier space, I get some products of the form:

##\displaystyle \left( \sum_{k=-\infty}^{\infty} \hat p_k e^{ikx} \right ) \left( -\sum_{k=-\infty}^{\infty} k^2 \hat u_k e^{ikx} \right)## and

##\displaystyle \left( \sum_{k=-\infty}^{\infty} ik \hat p_k e^{ikx} \right) \left( \sum_{k=-\infty}^{\infty} ik \hat u_k e^{ikx} \right)##,

So, my idea was (which worked for the constant coefficients situation) then project with ##\int_0^{2\pi}e^{-ik'x}dx## in the equation, and use the orthogonality relation ##\int_0^{2\pi} e^{ikx}e^{-ik'x}dx=2\pi \delta_{k,k'}## in order to get an equation for the ##\hat u_k##. However, in this situation it is not clear to me what should I get from:

##\displaystyle \int_0^{2\pi}e^{-ik'x}dx \left( \sum_{k=-\infty}^{\infty} \hat p_k e^{ikx} \right ) \left( -\sum_{k=-\infty}^{\infty} k^2 \hat u_k e^{ikx} \right)##

because of the product of sums. So, I'm not sure if this is a trivial question and I am not seeing it, or if I should look for another approach.

Any comments and ideas are welcome, thanks in advance.

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# I Heat equation with variable coefficients

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