It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features.


My question: Which Ansätze do you know to solve PDEs with Wavelets? Are these solution methods actually superior to the classical Ansätze?


Are there even Ansätze to solve stochastic partial differential equations? I am also especially interested in parabolic equations like e.g. diffusion equations.



Ansätze for solving PDEs with wavelets
The method of choosing a solution Ansatz to an equation and then actually deriving an exact solution is quite common in soliton theory, which is a sub-field of the study of hyperbolic equations. All methods described below, to my knowledge, only work on hyperbolic equations. Sorry, diffusion folks.


You must know properties of your equations to know which Ansatz will yield reasonable or good results. If you know that the tails of the solution die off quickly, you may choose a Gaussian $$ A \exp(-b x^2) $$, or if they die off **very** quickly, a super Gaussian $$ A \exp(-b a(x)^2) $$, where a(x) can be any polynomial. Also, based on the properties of your equation, you may want to multiply these 'basic' Ansatzen by other functions, to represent behavior that is known to be present. For example, if you know that solutions to the equation are not monotonic and/or 'wiggly', then you might want
$$ A \exp(-b x^2) \sin(k x) $$ The latter Ansatz is a two-parameter Ansatz and is the most likely to have a chance of working on a real equation. You may think that $ k $ is a third parameter, but actually, it is determined, usually algebraicly, by $A$ and $b$. Single parameter Ansatzen usually only work on very specific coefficients of equations and are too simple to model real equations.


There are obviously many, many other good Ansatzen, such as soliton solutions 
$$ A\ {\rm sech}^n{\left(k x - \omega t\right)} $$ (where $n$ is a positive even integer, and $\omega=\omega\left(k\right)$ is the dispersion relation)
if your equations has symmetry properties. There is a large theory, mostly derived from the work of R. Hirota, of how to derive exact solutions to systems of nonlinear PDE's which have certain symmetry properties or invariants, using the properties of bilinear operators.


Note: Directly translated, the word der Ansatz in German has many meanings, but it most usually is translated as 'approach' or 'basic approach', but it really just means: an educated guess of a solution, with enough degrees of freedom (in the form of parameters) such that the Ansatz is able to solve the equation.


Also, in the above equations, $A$ can be constant, or only a function of $ t $ or a function of both $x$ and $t$, depending on which behavior is being modeled.


