I have a query regarding the practical applicability of variable separable method. Usually we decompose the variables w.r.t components corresponding to different dimensions, this method is used to solve a differential equation analytically. When it comes to the practical usage, we are often provided already with the solution in form of model output. Is it possible to decompose them along their dimensions? For example I can write the zonal velocity as
U(x, y, z, t) = uv(z) * uh(x, y, t)
Will this decomposition be unique? I am clueless because it just seems to be a matrix multiplication. I wonder how would be the form of those matrices. Is it possible to obtain uv and uh from the data variable using any software tool?
Edit: An example of such a decomposition is provided here at page number 7 (equation 2.6). I wonder if it is possible to obtain the decomposed RHS values practically from a variable.
Moreover I often see a PDE associated with the decomposition which is absolutely logical. The equation is solved analytically by substituting the decomposed variables and different software tools are able to plot the analytical solution. The situation is little different in the study attached herewith. I wonder how to obtain the decomposed values if I am not getting any way to solve it analytically.