In this paper, we derive a solution to the telegrapher equation. We then apply a bifurcation parameter to the telegrapher equation in order to analyze the behavior of the solution as it changes classification. In order to obtain the solution to both the telegrapher and modified telegrapher equation, we derive the heat equation and telegrapher equation using a continuous random walk. We also solve the heat equation using invariant properties of a particular solution, a random walk analysis, and a Fourier-Laplace transform. The solution to the telegrapher equation contains modified Bessel functions, so we also derive the solutions to both the Bessel and modified Bessel equation. Lastly, we rigorously obtain a solution to the telegrapher equation with an added bifurcation parameter. This solution represents a complete distribution of the solution to the standard telegrapher equation as its solutions transition between classifications.
Wages, Hunter R., "An Analysis of the Telegrapher Equation with a Bifurcation Parameter to Model Relativistic Diffusion" (2019). Student Scholarship. 24.