In this article, we present the numerical solution of fractional Sturm-Liouville problems by using generalized shifted Chebyshev polynomials.
We combine right Caputo and left Riemann–Liouville fractional differential operators for the construction of fractional Sturm–Liouville operators. The proposed algorithm is developed using operational integration matrices of generalized shifted Chebyshev polynomials. We introduce a new bound on the coefficients of the shifted. Chebyshev polynomials subsequently employed to establish an upper bound for error in the approximation of a function by shifted Chebyshev polynomials.
We have solved fractional initial value problems, terminal value problems and Sturm–Liouville problems by plotting graphs and comparing the results. We have presented the comparison of approximated solutions with existing results and exact numerical solutions. The presented numerical problems with satisfactory results show the applicability of the proposed method to produce an approximate solution with accuracy.
The presented method has been applied to a specific class of fractional differential equations, which involve fractional derivatives of a function with respect to some other function. Keeping this in mind, we have modified the classical Chebyshev polynomials so that they involve the same function with respect to which fractional differentiation is performed. This modification is of great help to analyze the newly introduced polynomials from analytical and numerical point of view. We have compared our numerical results with some other numerical methods in the literature and obtained better results.
