Solving PDE: Is There a General Method or Just Guesswork?

[ksoy]

[SOLVED] Solving PDE

I am just wondering, is there any gerneral method in solving PDE's or just by guess works??

thanks...

 

[dextercioby]

I am just wondering, is there any gerneral method in solving PDE's or just by guess works??

thanks...

In mathematics there's no room for "guessing".
As for PDE-s,well,mathematicians found a way to classificate them.So far (and more than surely in the future) there has't been found a general method to solving PDE-s,that is to aapply successfully for every kind of PDE.
For example,for nonlinear PDE-s,there is no general method of solving.Analitically,of course.I assume that was the initial question about.
Try to solve (or imagine a way to tackling) somthing like that
$$\frac{u^{3}(x,y,z)}{xy^{\frac{6}{3}}z}[\frac{\partial^{5} u(x,y,z)}{\partial x^{5}}]^{7}+5 u^{8}(x,y,z)-12x^{7}y^{\frac{3}{4}}z=0$$

 

[ksoy]

how about linear PDE??
Is there a general method of solving them??

 

[dextercioby]

how about linear PDE??
Is there a general method of solving them??

Yes,for the linear case,there is.Try first of all to bring them to the canonical form.From there analyze the type (hyperboli,elliptic,parabolic) for every point in the domain of the unknown function.Then look very carefully at the geberal problem and its conditions (boundary type (Dirichlet/Neumann),or initial). Several methods come up then.Green function methods,variable separation methods,Fourier/Laplace transform methods,and so on.
Some equation,after being put in the canonical form may admit immediate integration,and the famous example is the unidimensional wave equation.

Anyway,all these depend from case to case.