Differential equations MOC

Similarity solutions

Similarity solutions is a technique for finding the solutions to a PDE by first observing symmetries of the PDE, that is transformations to dependent and independent variables under which the PDE is invariant. These symmetries are then exploited to produce similarity variables, which can be used to convert the PDE to an ODE.

Symmetries

Dilatational symmetry

Perhaps the simplest kind of symmetry is dilatational. Consider a PDE with with independent variables and dependent variable , e.g.

We apply the transformation

where and are to be determined (although one may turn out to be free). In this example,

In order for this to still be a solution, the coëfficients must be equal

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