Numerical solution of time-fractional telegraph equations using wavelet transform

This study solves the time-fractional telegraph equations with Dirichlet boundary conditions using a novel and effective wavelet collocation method based on Taylor wavelets. In the Caputo sense, the fractional derivative idea is used. The Taylor wavelets are created from the Taylor polynomials. The operational matrices of integration are extracted from the Taylor wavelets. The Taylor wavelet method (TWM) is developed using these operational integration matrices. According to this method, the selected telegraph equation is converted into a system of algebraic equations. Newton's iterative technique solves the attained system of algebraic equations. The proposed method's error estimate is provided. The projected method solution is compared with the numerical solutions of Sinc-Legendre, Legendre, and Fibonacci wavelet collocation methods in terms of the tables and graphs. The results obtained from the TWM are accurate and efficient. As we know, many PDEs do not have exact solutions, and some semi-analytical methods work based on controlling parameters, but this technique is free from controlling parameters. Also, it is easy to implement and consumes less time to execute the programs. The suggested wavelet-based numerical method is computationally appealing, successful, trustworthy, and resilient.