Fractional Telegraph Equation with the Caputo Derivative

The Cauchy problem for the telegraph equation (D-t(?))(2)u(t) + 2aD(t)(?)u(t) + Au(t) = f (t) (0 < t = T, 0 < ? < 1, a > 0), with the Caputo derivative is considered. Here, A is a selfadjoint positive operator, acting in a Hilbert space, H; D(t )is the Caputo fractional derivative. Conditions are found for the initial functions and the right side of the equation that guarantee both the existence and uniqueness of the solution of the Cauchy problem. It should be emphasized that these conditions turned out to be less restrictive than expected in a well-known paper by R. Cascaval et al. where a similar problem for a homogeneous equation with some restriction on the spectrum of the operator, A, was considered. We also prove stability estimates important for the application.

Automated summary

The Cauchy problem for the telegraph equation with the Caputo derivative and a selfadjoint positive operator is considered. Conditions are found for the existence and uniqueness of the solution. These conditions are less restrictive than expected, as shown in a previous study. Stability estimates important for application are also proven.