Open AccessA numerical study of fractional population growth and nuclear decay model
Author(s) -
Sara Salem Alzaid,
Pawan Kumar Shaw,
Sunil Kumar
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
ISSN - 2473-6988
DOI - 10.3934/math.2022637
Subject(s) - mathematics , ode , convergence (economics) , runge–kutta methods , ordinary differential equation , fractional calculus , linearization , stability (learning theory) , euler's formula , euler method , scheme (mathematics) , quadratic equation , order (exchange) , backward euler method , initial value problem , mathematical analysis , differential equation , euler equations , nonlinear system , computer science , physics , geometry , finance , quantum mechanics , machine learning , economic growth , economics
This paper is devoted to solving the initial value problem (IVP) of the fractional differential equation (FDE) in Caputo sense for arbitrary order $ \beta\in(0, 1] $. Based on a few examples and application models, the main motivation is to show that FDE may model more effectively than the ordinary differential equation (ODE). Here, two cubic convergence numerical schemes are developed: the fractional third-order Runge-Kutta (RK3) scheme and fractional strong stability preserving third-order Runge-Kutta (SSRK3) scheme. The approximated solution is derived without taking any assumption of perturbations and linearization. The schemes are presented, and the convergence of the schemes is established. Also, a comparative study has been done of our proposed scheme with fractional Euler method (EM) and fractional improved Euler method (IEM), which has linear and quadratic convergence rates, respectively. Illustrative examples and application examples with the numerical comparison between the proposed scheme, the exact solution, EM, and IEM are given to reveal our scheme's accuracy and efficiency.