Solving the n-term time fractional diffusion-wave problem via a method of approximation of tempered convolution

We present a method of approximation of tempered convolution via Laguerre polynomials and apply it in solving the n-term time fractional diffusion-wave problem in one spatial variable. In particular, we give a solution to the two-term diffusion-wave equation as an example of how the method works. The method is applicable in solving different fractional equations with entire and fractional derivatives in 2D. By a fractional operator, we transform them into convolution form and apply the approximation of tempered convolution by Laguerre polynomials. This leads to an infinite system of algebraic equations through coefficients whose solution gives the coefficients in the Laguerre series expansion of the solution to the corresponding equation.