NEW PREDICTOR-CORRECTOR APPROACH FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS: ERROR ANALYSIS AND STABILITY

Mohammad Shahbazi Asl; Mohammad Javidi; Bashir Ahmad

doi:10.11948/2156-907X.20180309

2019 Volume 9 Issue 4

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Mohammad Shahbazi Asl, Mohammad Javidi, Bashir Ahmad. NEW PREDICTOR-CORRECTOR APPROACH FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS: ERROR ANALYSIS AND STABILITY[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1527-1557. doi: 10.11948/2156-907X.20180309

Citation:

Mohammad Shahbazi Asl, Mohammad Javidi, Bashir Ahmad. NEW PREDICTOR-CORRECTOR APPROACH FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS: ERROR ANALYSIS AND STABILITY[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1527-1557. doi: 10.11948/2156-907X.20180309

NEW PREDICTOR-CORRECTOR APPROACH FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS: ERROR ANALYSIS AND STABILITY

1.
Department of Mathematics, University of Tabriz, Tabriz, Iran
2.
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Corresponding author: Email address: bashirahmad qau@yahoo.com (B.Ahmad)
Fund Project: This project is supported by a research grant of the University of Tabriz

Abstract

Abstract

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson's 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.
- Predictor-corrector approach /
- fractional differential equation /
- Caputo derivative /
- Volterra integral equation
MSC: 26A33, 45D05, 65L07, 65L20

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