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.
Highlights
The interest in Riemann’s method is long-standing
The reason is that once the Riemann function is determined, the governing equation can be solved for Cauchy data on any other non-characteristic curve
Copson [24] wrote the first review of Riemann’s method in 1958. He listed six different techniques to solve for the Riemann function
Summary
The reason is that once the Riemann function is determined, the governing equation can be solved for Cauchy data on any other non-characteristic curve. The value of such a property means that Riemann’s method continues to draw the attention of investigators today. Copson [24] wrote the first review of Riemann’s method in 1958 He listed six different techniques to solve for the Riemann function. R(r, s, r0 , s0 ) is called the Riemann function It is the solution of the characteristic boundary value problem for the adjoint equation denoted by (3)–(6).
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