### Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions

Firdous A. Shah, Mohd Irfan, Kottakkaran S. Nisar, R.T. Matoog, Emad E. Mahmoud
2021 Results in Physics
A B S T R A C T In this article, a new and efficient operational matrix method based on the amalgamation of Fibonacci wavelets and block pulse functions is proposed for the solutions of time-fractional telegraph equations with Dirichlet boundary conditions. The Fibonacci polynomials and the corresponding wavelets along with their fundamental properties are briefly studied at first. These functions along with their nice characteristics are then utilized to formulate the Fibonacci wavelet
more » ... nal matrices of fractional integrals. The proposed method reduces the fractional model into a system of algebraic equations, which can be solved using the classical Newton iteration method. Approximate solutions of the time-fractional telegraph equation are compared with the recently appeared Legendre and Sinc-Legendre wavelet collocation methods. The numerical outcomes show that the Fibonacci technique yields precise outcomes and is computationally more effective than the current ones. (E.E. Mahmoud). for < 0. Many phenomena in physical, chemical and biological sciences can be governed by the telegraph equations such as wave propagation in cable transmissions [1], acoustic waves in porous media [2], parallel flows in Maxwell fluids [3], population dynamics [4], pulsatile blood flow in arteries [5], and hyperbolic heat transfer [6]. Over a couple of decades, several researchers have solved the class of telegraph Eq. (1) by different analytical and numerical methods. For instance, finite difference scheme [7], variational iteration method [8], Chebyshev tau method [9], homotopy analysis method [10], Taylor matrix method [11], differential quadrature algorithm [12], -spline method [13], radial basis functions method [14], reduced differential transform method [15] and Bernoulli matrix method [16]. On the flipside, fractional calculus has captivated the scientific community because of its far and wide applications in physical, chemical, and biological sciences. The fractional derivative is a generalization of the integer-order derivative. The prime feature of the fractionalorder models is their non-local nature, due to which they can represent common physical processes and dynamical systems more precisely than the conventional integer-order models [17, 18] .