There is a typo at the end of the sentence starting with "Our goal is to step forward ..."
The term "nice enough" should be better explained.
Correct the typo "and \(" in the first line of "Fourier spectral methods for time-dependent PDEs". In the 5th line of "Galerkin methods:" starting with "the coefficients are defined.." exp(ijx) should be exp(-ijx) in the second term. Complete the citation of "Recommended reading".\)