AN EFFECTIVE METHOD FOR SOLVING THE MULTI TIME-FRACTIONAL TELEGRAPH EQUATION OF DISTRIBUTED ORDER BASED ON THE FRACTIONAL ORDER GEGENBAUER WAVELET

We investigate an effective method for solving the multi-time fractional telegraph equation of distributed order, combining the Regularized Beta function with the fractional- order Gegenbauer wavelet. In the first stage, we define the fractional-order Gegenbauer wavelet and then approximate the solution using this wavelet. We present an exact formula that incorporates the Regularized Beta function to compute the Riemann-Liouville fractional integral of this wavelet. The wavelet, along with the exact formula, is then applied to derive numerical solutions for the multidimensional time-fractional telegraph equation of distributed order. Utilizing the midpoint rule for the distributed integral term, we transform the fractional equation of distributed order into a multi-term fractional time-differential equation. The fractional derivative is employed in the Caputo sense, allowing us to reduce the numerical solutions of the multidimensional time-fractional telegraph equations to a system of algebraic equations. We provide an in-depth analysis of the convergence and error bounds of the proposed method. The applicability and efficiency of this methodology are demonstrated through four illustrative examples. Additionally, a comparison with existing results highlights the advantages of our numerical approach.