Numerical method for solving diffusion‐wave phenomena

The method is applicable for nonlinear problems too. We find solutions for the diffusion‐wave problem in $1D$ with $n$ ‐term time fractional derivatives whose orders belong to the intervals $(0,1),(1,2)$ and $(0,2)$ respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. This method transfers the diffusion‐wave problem into the corresponding infinite system of linear algebraic equations through the coefficients, which are uniquely solvable under some relations between the coefficients with index zero.