Bell wavelet-based numerical algorithm for fractional-order (1+1)-dimensional telegraph equations involving derivative in Caputo sense

This study presents an efficient numerical approach for solving fractional-order (1+1)-dimensional telegraph equations using a collocation method based on Bell wavelets. The method begins by exploring the fundamental properties of Bell polynomials and their associated wavelets. Fractional-order operational matrices of integrals (OMI) are developed using block-pulse functions, and the Bell wavelet OMI is then combined with collocation points to transform the telegraph equation into a system of algebraic equations. Newton's iterative technique is used to solve these equations. To validate the effectiveness and applicability of the proposed Bell wavelet method, several test problems are analyzed, with detailed error analysis included. Comparative results between the Bell wavelet approximate solutions and those from other established methods in the existing literature are presented through comprehensive figures and tables. This comparison highlights the improved accuracy and efficiency of the Bell wavelet method, demonstrating its potential as a reliable tool for solving fractional telegraph equations.