Partial differential equations (PDEs) describe a wide array of physical processes such has heat conduction and sound propagation. In many such equations, there are underlying "basic waves" that can be used to give efficient algorithms for computing solutions to these PDEs.

In all cases, one is provided with input data (such as a description of the medium and initial condition for a sound propagation experiment) and a specific PDE governing the physical process. One then writes the input data as a superposition of the basic waves of the PDE. In this form, the solution of the PDE is easily calculated.

*See also:* Finite element method Discrete element method

Here we presume a basic understanding of basic multivariate calculus and Fourier series. If g(x,y) is a complex-valued function of two real variables, and g is periodic in x and y (that is, g(x,y)=g(x+2π,y)=g(x,y+2π)) then we are interested in finding a function f(x,y) so that

- f
_{xx}(x,y)+f_{yy}(x,y)=g(x,y) for all x,y

If we write f and g in Fourier series:

- f=∑a
_{j,k}exp(ijx)exp(iky) - g=∑b
_{j,k}exp(ijx)exp(iky)

- ∑a
_{j,k}(-(j^{2}+k^{2}))exp(ijx)exp(iky)=∑b_{j,k}exp(ijx)exp(iky)

(*) a_{j,k}=-b_{j,k}/(j^{2}+k^{2})

which is an explicit formula for the Fourier coefficients a_{j,k}.

To turn this into an algorithm, only finitely many frequencies are solved for.

- Compute the Fourier transform (b
_{j,k}) of g. - Computer the Fourier transform (a
_{j,k}) of f via the formula (*) and the Fourier transform of g. - Compute f by taking an inverse Fourier transform of (a
_{j,k}).