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On the Riemann Function

Author

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  • Peter J. Zeitsch

    (School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia)

Abstract

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann’s method was published by E.T. Copson in 1958. This study extends that work. Firstly, three solution methods were overlooked in Copson’s original paper. Secondly, several new approaches for finding Riemann functions have been developed since 1958. Those techniques are included here and placed in the context of Copson’s original study. There are also numerous equivalences between Riemann functions that have not previously been identified in the literature. Those links are clarified here by showing that many known Riemann functions are often equivalent due to the governing equation admitting a symmetry algebra isomorphic to S L ( 2 , R ) . Alternatively, the equation admits a Lie-Bäcklund symmetry algebra. Combining the results from several methods, a new class of Riemann functions is then derived which admits no symmetries whatsoever.

Suggested Citation

  • Peter J. Zeitsch, 2018. "On the Riemann Function," Mathematics, MDPI, vol. 6(12), pages 1-31, December.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:12:p:316-:d:189392
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    References listed on IDEAS

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    1. Lubomir Dechevski & Nedyu Popivanov & Todor Popov, 2012. "Exact Asymptotic Expansion of Singular Solutions for the ( )-D Protter Problem," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-33, September.
    2. Nedyu Popivanov & Todor Popov & Allen Tesdall, 2014. "Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-19, October.
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