A Novel and Effective Scheme for Solving the Fractional Telegraph Problem via the Spectral Element Method

The combination of fractional derivatives (due to their global behavior) and the challenges related to hyperbolic PDEs pose formidable obstacles in solving fractional hyperbolic equations. Due to the importance and applications of the fractional telegraph equation, solving it and presenting accurate solutions via a novel and effective method can be useful. This work introduces and implements a method based on the spectral element method (SEM) that relies on interpolating scaling functions (ISFs). Through the use of an orthonormal projection, the method maps the equation to scaling spaces raised from multi-resolution analysis (MRA). To achieve this, the Caputo fractional derivative (CFD) is represented by ISFs as a square matrix. Remarkable efficiency, ease of implementation, and precision are the distinguishing features of the presented method. An analysis is provided to demonstrate the convergence of the scheme, and illustrative examples validate our method.