Approximate solution of fractional vibration equation using Jacobi polynomials

An approximate method using Jacobi polynomials is proposed for the approximate solutions of fractional vibration equation.Convergence and numerical stability of the proposed method are shown.Results are discussed for different cases of Jacobi polynomials.Results are compared with existing analytical methods. In this paper, we present a method based on the Jacobi polynomials for the approximate solution to fractional vibration equation (FVE) of large membranes. Proposed method converts the FVE into Sylvester form of algebraic equations, whose solution gives the approximate solution. Convergence analysis of the proposed method is given. It is also shown that our approximate method is numerically stable. Numerical results are discussed for different values of wave velocities and fractional order involved in the FVE. These numerical results are shown through figures for particular cases of Jacobi polynomials such as (1) Legendre polynomial, (2) Chebyshev polynomial of second kind, (3) Chebyshev polynomial of third kind, (4) Chebyshev polynomial of fourth kind, (5) Gegenbauer polynomial. The accuracy of the proposed method is proved by comparing results of our method and other exiting analytical methods. Comparison of results are presented in the form of tables for particular cases of FVE and Jacobi polynomials.