Abstract
In this paper, a comparative study has been made between different algorithms to find the numerical solutions of the fractional-order clock chemical model (FOCCM). The spectral collocation method (SCM) with the shifted Legendre polynomials, the two-stage fractional Runge–Kutta method (TSFRK) and the four-stage fractional Runge–Kutta method (FSFRK) are used to approximate the numerical solutions of FOCCM. Our results are compared with the results obtained for the numerical solutions that are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation (LPI). Firstly, the accuracy of the results is checked by computing the absolute error between the numerical solutions by using SCM, TSFRK, FSFRK, and LPI and the exact solution in the case of the fractional-order logistic equation (FOLE). The numerical results demonstrate the accuracy of the proposed method. It is observed that the FSFRK is better than those by SCM, TSFRK and LPI in the case of an integer order. However, the non-integer orders in the cases of the SCM and LPI are better than those obtained by using the TSFRK and FSFRK. Secondly, the absolute error between the numerical solutions of FOCCM based upon SCM, TSFFRK, FSFRK, and LPI for integer order and non-integer order has been computed. The absolute error in the case of the integer order by using the three methods of the third order is considered. For the non-integer order, the order of the absolute error in the case of SCM is found to be the best. Finally, these results are graphically illustrated by means of different figures.
Highlights
We first conduct a test for all methods on fractional-order Logistic equation, which has an exact solution for integer as well as non-integer order
In order to verify the efficiency of the results, which we have investigated in this paper, we introduced the fractional-order logistic equation with its exact solution
We compared the numerical solutions based on the spectral collocation method (SCM), two-stage fractional Runge–Kutta method (TSFRK), four-stage fractional Runge–Kutta method (FSFRK), and Lagrange polynomial interpolation (LPI), respectively, with the exact solution of the fractional-order logistic equation for integer and non integer orders
Summary
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40204, Taiwan Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz P.O. Box 6803, Yemen
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