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As far as I understand, both the method of characteristics and similarity solutions allow us to reduce certain partial differential equations to ordinary differential equations which can then be solved.

Is the underlying mathematical 'machinery' the same for both methods?

Any explanation of this, or book recommendations would be greatly appreciated. My background is in engineering (with a dash of applied mathematics) so that may restrict me from understanding more pure expositions on the subject. Thanks.

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2 Answers 2

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The method of characteristics is appropriate to solve initial value problems of hyperbolic type: semi linear first order differential equations, one-dimensional wave equation. In principle all solutions can be found using this method.

Similarity solutions are a special type of solutions that reflect invariant properties of the equation. Very often they have a special significance.

Another type of special solution is the so called traveling wave solutions, that is, solutions of the form $\phi(x-c\,t)$, representing a wave of shape $\phi$ traveling with speed $c$.

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  • $\begingroup$ Really interesting thanks. However, I have a further question. Does the method of characteristics also reflect an invariant property of the equation in question, along the so-called 'characteristic' lines? $\endgroup$
    – Moustache
    Commented Dec 13, 2016 at 11:52
  • $\begingroup$ Characteristic lines reflect properties of the equation that may not be directly related to invariance. For instance, singularities in wave equations propagate along characteristics. $\endgroup$ Commented Dec 13, 2016 at 12:05
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Just adding a couple of points to Julian's answer: In case of a physical problem, it is worthwhile trying similarity solutions whenever a problem has no geometric length-scale associated with it (i.e. you are working in an infinite domain). E.g. Heating of an infinite slab.

Method of characteristics is applicable mostly for hyperbolic problems. Higher order PDEs can also be sometimes solved by casting them as a system of hyperbolic PDEs. But most often, the idea of characteristics is used to obtain an intuition for the problem in hand and the kind of boundary or initial conditions one would need to specify for numerical approaches.

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