The g-analytic Function Theory and Wave Equation

  •  Chein-Shan Liu    


In this paper we develop a $g$-analytic function and a $g$-harmonic function theory for one-dimensional wave equation in the Minkowski space. In terms of the Minkowskian polar coordinates we can derive a set of complete hyperbolic type Trefftz bases, which can be transformed to polynomials as the bases for a trial solution of wave equation. The Cauchy-Riemann equations and the Cauchy theoremfor $g$-analytic functions are proved, and meanwhilethe existence of Cauchy integral formula is disproved from thenon-uniqueness of the Dirichlet problem for wave equation under the boundary conditions on whole boundary, which isalso known as the backward wave problem (BWP).Examples are used to demonstrate these results.

This work is licensed under a Creative Commons Attribution 4.0 License.