Towards a generalization of the separation of variables technique

The method of separation of variables is simple, elegant and very powerful but has been applied to a limited number of differential operators both linear as well as non-linear. The underlying reason can be sought in the common belief that separation of variables for higher-order partial differential equations which include mixed derivatives is not possible. Although, the statement is valid when separation of variables is applied in its traditional form, these impediments can be bypassed introducing a generalized version of this over 250 years old technique. This will be attempted in the context of the present article. After familiarizing the reader with the concepts of the generalized form of the method of reduction, emphasis is placed on the effectiveness of the technique, providing explicit solutions to higher-order linear partial differential equations incorporating mixed derivatives.

Keywords

generalized separation of variables, $n$–harmonic equation, $n$–Helmholtz equation, $n$–metaharmonic equation

2010 Mathematics Subject Classification

31B30, 33E30

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Published 17 April 2013