Integral decomposition for the solutions of the generalized Cattaneo equation

We present the integral decomposition for fundamental solution of generalized Cattaneo equation with both time derivatives smeared through convoluting them some memory kernels. For power-law kernels ${t}^{\ensuremath{-}\ensuremath{\alpha}}, \ensuremath{\alpha}\ensuremath{\in}(0,1]$ this becomes fractional one governed by Caputo in which highest order is 2. To invert solutions from Fourier-Laplace domain to space-time we use analytic methods based on Efross theorem and find out that looked are represented decompositions tangle standard nonnegative normalizable functions being uniquely dependent Furthermore, methodology arising theory complete Bernstein allows us assign such constructed interpretation subordination. This fact preserved two limit cases built into equations, i.e., either diffusion or wave equations. point applying enables go beyond approach usually leads involving Gaussian distribution describing Brownian motion. Our clarifies puzzling situation takes place ${t}^{\ensuremath{-}\ensuremath{\alpha}}$ subordination motion does not work if $\ensuremath{\alpha}\ensuremath{\in}(1/2,1]$.