Fractional telegraph equations.
The main result of this paper is the following. Let \(A=A^{*}\) be a positive self-adjoint operator on a Hilbert space \(\mathcal H\) and let \(a\) be a positive constant. Let \(f_1\) and \(f_2\) be arbitrary and let \(u=u(t)\) be the unique solution of the initial value problem \[ (D^{\gamma})^2u+2aD^{\gamma}u+Au=0, \quad u(0)=f_1,\quad D^{\gamma}u(0)=f_2, \quad0<\gamma<1. \] Then there exists \(v=v(t)\), a solution of \(2aD^{\gamma}v+Av=0\), which has the same asymptotic behavior as \(u\), in the sense that \(u(t)=v(t)+o(v(t))\) as \(t\to+\infty\). Here the convergence is in a weak sense.
scientific article; zbMATH DE number 1866737