Abstract
We describe how the construction of similarity solutions of partial differential equations extends naturally from concepts in dimensional analysis. In particular, we show how to obtain self-similar solutions through scaling invariances of linear and nonlinear PDE. We give examples illustrating how similarity solutions of PDEs can be obtained from solutions of ODE problems.
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Notes
- 1.
The choice \(b=1\) leads to a differential equation for the similarity function whose dependence on \(\eta \)-derivatives will mirror the x-derivatives in the original problem.
- 2.
Without loss of generality, we assume the support to be symmetric relative to the origin.
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© 2015 Springer International Publishing Switzerland
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Witelski, T., Bowen, M. (2015). Self-Similar Scaling Solutions of Differential Equations. In: Methods of Mathematical Modelling. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23042-9_5
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DOI: https://doi.org/10.1007/978-3-319-23042-9_5
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23041-2
Online ISBN: 978-3-319-23042-9
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