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I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way:

$\hat{u}_{t}-(i\omega)^{2}\hat{u}_{t}=\hat{F}(u)$ where $F(u)=bu_{x}u_{xx}-(b+1)uu_{x}+uu_{xxx}$

$\hat{u}_{t}=\frac{\hat{F}(u)}{1+\omega^{2}}$ and so on.. Now I'm trying to solve the following PDE:

$u_{t}=\frac{1}{4}u_{x}^{2}u_{t}-\frac{1}{4}uu_{xx}u_{t}-\frac{1}{4}uu_{x}(u_{t})_{x}+\frac{1}{4}u^{2}(u_{t})_{xx}-u^{3}u_{x}$.

My question is if there a way to bring this equation, by using some manipulations, to the form : $\hat{u}_{t}=\hat{F}(u)$ where $F(u)$ is some combination of functions which don't involve partial derivatives of $u$ in respect to t?

Thanks a lot!

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