Spectral method: Difference between revisions

Revision as of 13:13, 27 September 2018

Revision as of 08:07, 1 May 2019

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===A concrete, linear example===

Here we presume an understanding of basic multivariate [[calculus]] and [[Fourier series]]. If <math>g(x,y)</math> is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, <math>g(x,y)=g(x+2π2\pi,y)=g(x,y+2π2\pi)</math>) then we are interested in finding a function f(x,y) so that

:<math>\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)f(x,y)=g(x,y)\quad \text{for all } x,y</math>