Received: 15 October 2020 Revised: 30 October 2020 Accepted: 5 November 2020 DOI: 10.1002/num.22684 RESEARCH ARTICLE A numerical study on fractional differential equation with population growth model Sunil Kumar 1 Pawan Kumar Shaw 1 Abdel-Haleem Abdel-Aty 2,3 Emad E. Mahmoud 4,5 1 Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand, India 2 Department of Physics, College of Sciences, University of Bisha, Bisha, Saudi Arabia 3 Physics Department, Faculty of Science, Al-Azhar University, Assiut, Egypt 4 Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia 5 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt Correspondence Abdel-Haleem Abdel-Aty, Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia; Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt. Email: amabdelaty@ub.edu.sa Funding information Taif University, Grant/Award Number: TURSP-2020/20. Abstract In this work, we developed two efficient and fast numerical technique to solve an initial value problem (IVP) of the lin- ear and nonlinear fractional differential equations (FDEs) of order α,0 < α < 1. Here we have used the arbitrary order derivatives in Riemann style. The proposed algorithm are very accurate and provides the solutions directly with- out perturbations, linearization, or any other assumptions. Illustrating examples with numerical comparisons between the proposed algorithm and the exact and/or Euler method and the improved Euler method (IEM) are given to reveal the efficiency and the accuracy of our algorithm. These scheme has quadratic and cubic convergence rate which is faster than the Euler method and IEM for solving the IVP of FDEs. Moreover, we have discussed the behaviors through graphical representation of the obtained solutions. Furthermore, both methods will be useful for the treatment of disease models for further study. KEYWORDS Euler method, FDEs, Heun method, improved Euler method, initial value problem, midpoint method, popula- tion growth model 1 INTRODUCTION In our recent decades, the numerical treatment of arbitrary order differential equation is one of the most popular issue in applied fractional calculus. Leibniz was the first person who find the gap of calcu- lus and invented the concept of fractional calculus in the year 1695 [22, 40, 52] and further fractional Numer Methods Partial Differential Eq. 2020;1–22. wileyonlinelibrary.com/journal/num © 2020 Wiley Periodicals LLC 1