Approximate Solution of Mixed Problem for Telegrapher Equation with Homogeneous Boundary Conditions of First Kind Using Special Functions

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that line is free from distortions, reduced to a similar one-dimensional inhomogeneous wave equation. An effective way solve this based on use of special functions – polylogarithms, which are complex power series with coefficients, converging unit circle. exact solution expressed integral form terms imaginary part first-order polylogarithm circle, approximate one finite sum real dilogarithm third-order polylogarithm. All indicated parts polylogarithms periodic have polynomial expressions corresponding degrees an interval length period, makes it possible obtain elementary functions. In paper, we study telegrapher’s applications. This linear substitution desired function witha time-exponential coefficient Klein Gordon latter can be found by dividing variables trigonometric point time-dependent coefficients. Such little practical application, since requires calculation large number coefficients-integrals difficult estimate error calculations. present propose another problem, He-functions, certain type converge presented second-order He-functions final He-functions. paper also proposes simple problem. It relation splitting step coefficient. example solving has been developed given, graphs solutions constructed.