This paper presents an efficient and optimal numerical method for obtaining approximate solutions to the time-fractional telegraph equation in the distributed-order model as \(\begin{aligned} ^{C}D_{t}^{\alpha }z(x,t)+\int \limits _{1}^{2} {\mathcal {B}}(\beta ) ^CD_{t}^{\beta }z(x,t)\textrm{d}\beta +z(x,t)=\frac{\partial ^{2}}{\partial x^{2}}z(x,t)+y(x,t). \end{aligned}\) A hybrid approach is proposed, combining the midpoint method for approximating the distributed-order integral, the regularized beta function, and the fractional-order Gegenbauer wavelet method. An exact formula involving the regularized beta function is derived for computing the Riemann–Liouville fractional integral of this class of wavelets. The proposed wavelet, along with the exact formula, is applied to obtain numerical solutions for the multi-term time-fractional telegraph equation of distributed order. By employing the midpoint rule for the distributed integral term, the given fractional equation is transformed into a multi-term time-fractional differential equation in time. The fractional derivative is considered in the Caputo sense. This method effectively reduces the time-fractional telegraph equations to a system of algebraic equations. Convergence analysis and error bounds of the proposed approach are investigated. The applicability and efficiency of the method are demonstrated through two numerical examples. Furthermore, comparisons with existing results highlight the advantages of our numerical approach.
