Article Sidebar

Download
PDF
Statistic
Read Counter : 144 Download : 59

Main Article Content

Abstract

This paper investigates the accuracy of the Predictor–Corrector numerical method for solving linear differential equations with Caputo-type fractional derivatives. Fractional derivatives defined in the sense of Caputo are particularly important due to their ability to model memory and time-dependent behaviors in physical and engineering systems. The aim of this study is to evaluate the efficiency and accuracy of the Predictor–Corrector method in solving linear fractional differential equations. To achieve this, the structure of the method is first introduced, and then it is applied to several numerical examples, including both homogeneous and nonhomogeneous equations. The numerical results are compared with analytical solutions, and tables and comparative graphs are presented to assess the method's accuracy. The findings indicate that the Predictor–Corrector method provides reliable and accurate results and can be considered an effective approach for numerically solving linear fractional differential equations with classical initial conditions.

Keywords

Caputo Derivative Fractional Differential Equations Laplace Transform Numerical Approximation Predictor-Corrector Method

Article Details

License

Copyright (c) 2025 Reserved for Kabul University

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

How to Cite
Haidari, M. J., Noori, N., & Baidar, A. W. (2025). Accuracy Analysis of the Numerical Solution of Linear Fractional-Order Differential Equations of Caputo Type Using the Predictor-Corrector Method . Journal of Natural Sciences – Kabul University, 8(3), 31–56. https://doi.org/10.62810/jns.v8i3.439
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX

References

  1. Bagley, R. L., & Torvik, P. J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27(3), 201–210. http://dx.doi.org/10.1122/1.549724 DOI: https://doi.org/10.1122/1.549724
  2. Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent—II. Geophysical Journal International, 13(5), 529–539.https://doi.org/10.1111/j.1365-246X.1967.tb02303.x DOI: https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  3. Deng, W. (2007). Short memory principle and a predictor–corrector approach for fractional differential equations. Journal of Computational and Applied Mathematics, 206(1), 174–188. https://doi.org/10.1016/j.cam.2006.07.006 DOI: https://doi.org/10.1016/j.cam.2006.06.008
  4. Diethelm, K. (1997). An algorithm for the numerical solution of differential equations of fractional order. Electronic Transactions on Numerical Analysis, 5, 1–6. Link
  5. Diethelm, K., & Freed, A. D. (1999). On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In Scientific Computing in Chemical Engineering II (pp. 217–224). Springer. DOI: https://doi.org/10.1007/978-3-642-60185-9_24
  6. https://doi.org/10.1007/978-3-642-59953-5_21
  7. Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29, 3–22. https://doi.org/10.1023/A:1016592212823 DOI: https://doi.org/10.1023/A:1016592219341
  8. Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer. https://doi.org/10.1007/978-3-642-14574-2 DOI: https://doi.org/10.1007/978-3-642-14574-2
  9. Ford, N. J., & Simpson, A. C. (2001). The numerical solution of fractional differential equations: Speed versus accuracy. Numerical Algorithms, 26, 333–346. https://doi.org/10.1023/A:1016662221662 DOI: https://doi.org/10.1023/A:1016601312158
  10. Garrappa, R. (2010). On some explicit Adams-type predictor–corrector methods for fractional differential equations. Journal of Computational and Applied Mathematics, 229(2), 392–399. https://doi.org/10.1016/j.cam.2008.11.005 DOI: https://doi.org/10.1016/j.cam.2008.04.004
  11. Garrappa, R. (2015). Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics, 3(4), 337–384. https://doi.org/10.3390/math3040337
  12. Heydari, M. H., & Hooshmandasl, M. R. (2013). A new numerical method for solving nonlinear fractional differential equations using predictor–corrector scheme. Mathematical Problems in Engineering, 2013, 1–10. https://doi.org/10.1155/2013/184726 DOI: https://doi.org/10.1155/2013/312328
  13. Ishteva, M. (2005). Properties and applications of the Caputo fractional operator (Report No. 5). Department of Mathematics, University of Karlsruhe.
  14. Kimeu, J. M. (2009). Fractional calculus: Definitions and applications. Western Kentucky University. Link
  15. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier.
  16. https://doi.org/10.1016/S0076-5392(05)80002-0
  17. Li, C., & Zeng, F. (2012). The predictor–corrector approach for fractional differential equations revisited. Applied Mathematics Letters, 25(9), 1355–1360. https://doi.org/10.1016/j.aml.2012.05.022
  18. Li, C., & Zeng, F. (2015). Numerical methods for fractional calculus. CRC Press. DOI: https://doi.org/10.1201/b18503
  19. Li, C., Zeng, F., & Chen, Y. Q. (2015). A survey of numerical methods for fractional differential equations. Journal of Computational Physics, 293, 261–283. https://doi.org/10.1016/j.jcp.2015.01.013 DOI: https://doi.org/10.1016/j.jcp.2015.01.013
  20. Li, C., Zeng, F., & Liu, F. (2019). Numerical methods for fractional differential equations with nonlinear terms. Applied Mathematics Letters, 94, 149–155. https://doi.org/10.1016/j.aml.2019.02.015 DOI: https://doi.org/10.1016/j.aml.2019.02.015
  21. Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V., & Zhang, Y. (2021). Machine learning methods for solving fractional differential equations. Computers, Materials & Continua, 66(1), 537–552.
  22. https://doi.org/10.32604/cmc.2020.011167
  23. Lubich, C. (1986). Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3), 704–719. https://doi.org/10.1137/0517050 DOI: https://doi.org/10.1137/0517050
  24. Mainardi, F. (2010). Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models. World Scientific. https://doi.org/10.1142/7435 DOI: https://doi.org/10.1142/9781848163300
  25. Odibat, Z. M., & Shawagfeh, N. T. (2007). Generalized Taylor’s formula. Applied Mathematics and Computation, 186(1), 286–293. https://doi.org/10.1016/j.amc.2006.07.045 DOI: https://doi.org/10.1016/j.amc.2006.07.102
  26. Podlubny, I. (1999). Fractional differential equations. Academic Press.
  27. Sousa, E. (2016). A second order explicit finite difference method for the fractional advection diffusion equation. Computers & Mathematics with Applications, 73(5), 811–825. https://doi.org/10.1016/j.camwa.2016.01.016 DOI: https://doi.org/10.1016/j.camwa.2016.01.016
  28. Zabidi, N. A., Abdul Majid, Z., Kilicman, A., & Ibrahim, Z. B. (2022). Numerical solution of fractional differential equations with Caputo derivative by using numerical fractional predict–correct technique.
  29. https://doi.org/10.1186/s13662-022-03697-6 DOI: https://doi.org/10.1186/s13662-022-03697-6
  30. Zayernouri, M., & Karniadakis, G. E. (2013). Fractional Sturm–Liouville eigen-problems: Theory and numerical approximations. Journal of Computational Physics, 252, 495–517. https://doi.org/10.1016/j.jcp.2013.06.010 DOI: https://doi.org/10.1016/j.jcp.2013.06.031
  31. Zeng, F., Li, C., Liu, F., & Turner, I. (2018). A stable and high-order method for the time-fractional Black–Scholes equation. SIAM Journal on Scientific Computing, 40(5), A3046–A3069. https://doi.org/10.1137/17M1159648