A new efficient algorithm based on fifth-kind Chebyshev polynomials for solving multi-term variable-order time-fractional diffusion-wave equation

An algorithm based on a class of the Chebyshev polynomials family called the fifth-kind Chebyshev polynomials (FCPs) is introduced to solve the multi-term variable-order time-fractional diffusion-wave equation (MVTFD-WE). Appeared fractional derivative operators in these equations are of the Caputo type. Coupling FCPs and the collocation method leads to reduce the MVTFD-WE to a system of algebraic equations. The convergence of the proposed scheme is investigated in a weighted Sobolev space via obtaining error bounds for approximate solutions which shows the method error tends to zero if the number of terms of the series solution is selected sufficiently large. The applicability and efficiency of the suggested method are examined through several illustrative examples.