A highly accurate and efficient Genocchi-based spectral technique applied to singular fractional order boundary value problems

Abstract

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two-term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the $L^2$ norm is obtained via a rigorous error analysis. The main benefit of the presented QLM-GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high-order accuracy of the QLM-GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well-established available computational methods. It is apparently shown that the QLM-GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two-term fractional derivatives accurately and efficiently.