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I am wondering, is there a general method one uses to seek methods of solutions without much strenuous work and calculations, and possibly a way to get non-trivial solutions for PDE's.

A simple counter example would be:

Suppose we are given: $\dfrac{\partial u}{\partial x}-2u = 0$

And are asked to find solutions that would satisfy this equation. Well a sort of trivial solution would be to say, suppose we let $u=e^{2x}$. This would imply from the equation that, $$2e^{2x}-2e^{2x}=0.$$

Now we all know that this would not be the case for much tougher equations to begin with, especially non-linear equations. What I want to find out is, is there a systematic manner of sifting out the other solutions that are more non-trivial without a great deal of work. Maybe like some common trick or something. It is sort of like for ODE's of certain type, we can expect an exponential form of a solution.

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up vote 4 down vote accepted

There is no general rule to find solutions for PDE's. However, there are special types of solutions that may help when looking for particular solutions. In many cases, these special solutions are the ones of physical interest, or describe the cualitative behviour of the general solution. A non comprehensive list of special type of solutions follows. I asume a PDE with independent variables $t\in\mathbb{R}$ (time) and $x=(x_1,\dots,x_n)\in\mathbb{R}^n$ (space).

  • Stationary solutions: solutions that do not depend on time.
  • Solutions in separeted variables: solutions of the form $T(t)\,X(x)$. They are the base of the Fourier method for solving initial and boundary problems for linear PDE's.
  • Radial solutions: solutions with rotational symmetry of the form $u(r,t)$ where $r=|x|$.
  • Self similar solutions: solutions invariant under the action of a group of transformations of the form $T_\lambda u(t,x)=\lambda^au(\lambda^b\,x,\lambda^c\,t)$ for some $a,b,c\in\mathbb{R}$.
  • The previous two are particular examples of solutions invariant under the action of a Lie group. A reference is the book Applications of Lie Groups to Differential Equations by Peter J. Olver.
  • Travelling waves: solutions of the form $u=\phi(x-c\,t)$. They are solutions with a fixed profile that travels at speed $c$.
  • ...
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Hey Julian, This is a really well thought out response and helpful to know for future reference and work/research. Thank You. –  night owl Feb 18 '12 at 6:12
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