A lot of time i found different papers in which numerical methods execute to find the solution of given model these numerical methods apply with initial condition but in analytical methods we used wave transformation to convert PDE into ODE. Here my question is raised in any case there is numerical method in applied and wave transformation is not given and we want to apply the analytical method, analytical method need wave transformation so, how can we make a wave transformation?
What do you mean by "wave transformation"? Fourier transform, for example?
Dear Waseem
The Lie symmetry method can be applied to construct such transformations.
Regards,
Kamyar
@Roberto Ferretti
A transformation used in analytical methods to convert PDE into ODE
If you are on a d-dimensional interval, you can expand the solution in the Fourier basis, then rewrite the PDE in this basis. For example, the heat equation would turn into a system of decoupled linear ODEs, with coefficient given by the eigenvalues of the Laplacian. This idea is the basis for the so-called spectral methods (see for example the book by Canuto, Hussaini, Quarteroni and Zhang).
Numerically, the initial condition is transformed into the Fourier domain by Fast Fourier Transform (FFT), and the final solution is transformed back into the original domain by Inverse FFT.
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