Crude Fourier Series approximation for PDEs.

Is there a way to "crudely" approximate PDEs with Fourier series?

By saying crudely, I meant this way:

Assuming I want a crude value for a differential equation using Taylor series;

y' = x + y, y(0) = 1

i'd take a = 0 (since initially x = 0),

y(a) = 1,
y'(x) = x + y; y'(a) = 0 + 1 = 1
y"(x) = 1 + y'; y"(a) = 1 + 1 = 2
y'"(x) = 0 + y"(x); y"'(a) = 0 + 2 = 2

then y ~ 1 + x + 2/2! x^2 + 2/3! x^3.

Or something similar to that. Does this crude method have an analog to Fourier-PDE solutions?
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 Hey maistral. With a fourier series, you need to project your function to the fourier space to get the co-effecients. So the question is, how do you get an appropriate function to project to the trig basis if it's not explicit (i.e. you don't have f(x) but a DE system that describes it)?

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