A probability method for the solution of the telegraph equation with real-analytical initial conditions (Q2722142)

The solution of the Cauchy problem for the telegraph equation was obtained by B. Riemann. \textit{A. F. Turbin} [Fractal Analysis and Related Problems. Kyiv, No.2, 47-60 (1998)] proposed an algorithm for construction of a solution to the Cauchy problem for a special type of the initial conditions based on calculation of moments of the process of one-dimensional random motion \(\beta(t)=x+\nu\int^{t}_{0}(-1)_{\xi_{r}^{\lambda}}(s) ds,\) where \(\nu>0,\) \(\xi_{r}^{\lambda}(t)\) is a Markov chain with values in \(\{0,1\}.\) In the present work the authors develop the algorithm proposed by A. F. Turbin for the real-analytical initial conditions. The proposed probabilistic approach based on explicit calculations for arbitrary moments of the random process \(\beta(t)\) gives a possibility to avoid the analytical difficulties and to obtain new integral formulas for Bessel functions.