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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.

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The german word Ansatz (plural Ansaetze) is used in many different ways when appropriated into English in my experience. Do you mean "methods" or "theorems"? I've seen both, but perhaps I've just seen bad germlish (german mixed with english). Good question though, I'm rummaging around to see if I can find an answer! – Michael Hoffman Oct 30 '09 at 7:51
At least in this question it goes more into the direction of "methods". Thank you for your support! – vonjd Oct 30 '09 at 8:37
Oh, also, what do you mean by "superior"? BTW, you seem particularly interested in math-finance, have you checked out Steve Shreve's series? they may have some answers for a lot of your questions... – Michael Hoffman Oct 30 '09 at 13:39
Good question - I would think that they could produce more neatly arranged solutions because they are more localized compared to e.g. sin-waves that always go from -oo to oo. I know the works of Shreve, but I must admit that I still don't dare reading them because I deem them to sophisticated for me (you see, I am only an amateur fascinated by math!) – vonjd Oct 30 '09 at 14:12
up vote 1 down vote accepted

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 $$ A {sech}^n{\left(k x - \omega t\right)} $$ (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.

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If you mean numerical methods, writing your solution as a linear combination of wavelets and truncating the series in various ways gives you different methods. Particularly powerful one is Galerkin methods which are also a basis of finite element methods. I would say in practice finite element or other methods are better than wavelet methods for low dimensional problems. For high dimensional problems, and singular integral equations wavelets have advantage, especially the adaptive wavelet algorithms.

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