Wavelets method for the time fractional diffusion-wave equation

• A new operational matrix of fractional integration for the LWs is derived.
• A new method based on the LWs is proposed for the time FDWE.
• The paper contains some useful properties of the LWs.
• The proposed method can be applied for fractional sub-diffusion systems.
• The proposed method can be extended for fourth-order FDWE.

In this paper, an efficient and accurate computational method based on the Legendre wavelets (LWs) is proposed for solving the time fractional diffusion-wave equation (FDWE). To this end, a new fractional operational matrix (FOM) of integration for the LWs is derived. The LWs and their FOM of integration are used to transform the problem under consideration into a linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically.