I have a PDE that contains both the 3rd derivative and 4th derivative. Example shown below

$$ \frac{\partial u}{\partial t} =\frac{\partial}{\partial x}(u^3\frac{\partial^3u}{\partial x^3}) $$

$$ \frac{\partial u}{\partial t} =(u^3\frac{\partial^4u}{\partial x^4} + 3u^2\frac{\partial u}{\partial x}\frac{\partial^3u}{\partial x^3}) $$

From the material "Spectral Methods in Matlab", it says that for the second derivative it takes the Chebyshev differentiation matrix as $D^2$. Does that mean can I consider the Chebyshev differentiation matrix for the 3rd derivative and 4th derivative as $D^3$ and $D^4$ respectively? Is the below equation correct for me to use ode15s to solve it?

$$ \frac{\partial u}{\partial t} =(u^3*(D^4*u) + 3u^2*(D*u)*(D^3u)) $$

Trefethen, Lloyd N., Spectral methods in Matlab, Software - Environments - Tools. 10. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. 184 p. (2000). ZBL0953.68643.