Wave and telegraph equations with real time variable and complex spatial variables
The classical heat and Laplace equations with real time variable and complex spatial variable are studied. The purpose of this article is to make a similar study for the classical wave and telegraph equations with real time variable and complex spatial variable. The complexification of the spatial variable in the wave and telegraph equations is made by two different methods which produce different equations. By the former method, we complexify the spatial variable in the corresponding formulas by replacing the usual translations x ± ct, c is the speed of propagation, by the rotations ze ±ict and, by the latter, we complexify the spatial variable in the corresponding evolution equation and then we search for analytic and non-analytic solutions. The first method produces solutions that also preserve some geometric properties of the boundary function, such as the univalence, starlikeness, convexity and spirallikeness. Moreover, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated from both methods and their solutions are constructed. New physical/probabilistic interpretations of the solutions to these equations are also given. © 2011 Copyright Taylor and Francis Group, LLC.
Complex Variables and Elliptic Equations
Gal, C., Gal, S., & Goldstein, J. (2012). Wave and telegraph equations with real time variable and complex spatial variables. Complex Variables and Elliptic Equations, 57 (1), 91-109. https://doi.org/10.1080/17476931003786683