On a New Predictor-Corrector Scheme for Solving Nonlinear Differential Equations with Conformable Fractional Derivative Operator
Abstract
This work proposes a conformable fractional predictor-corrector algorithm for solving conformable fractional differential equations. Fractional calculus is finding applications in various scientific fields; therefore, developing numerical methods to solve fractional differential equations that model natural phenomena is of high importance, especially when finding analytical solutions is of high difficulty. Many authors have developed numerical methods for other fractional derivatives, such as the Caputo fractional derivative. In this article, our aim is to design Adams-Bashforth and Adams-Moulton methods specifically tailored for the conformable fractional derivative. Some examples are provided to illustrate the applicability of the numerical method.
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References
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[2] J. T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Communications in nonlinear science and numerical simulation, 16(3), 1140–1153, (2011).
[3] A. S. Shaikh, I. N. Shaikh and K. S. Nisar, A mathematical model of covid-19 using
fractional derivative: outbreak in india with dynamics of transmission and control,
Advances in Difference Equations, 2020(1), 373, (2020).
[4] R. L. Magin, Fractional calculus models of complex dynamics in biological tissues,
Computers & Mathematics with Applications, 59(5), 1586–1593, (2010).
[5] M. Dalir and M. Bashour, Applications of fractional calculus, Applied Mathematical
Sciences, 4(21), 1021–1032, (2010).
[6] C. G. Koh and J. M. Kelly, Application of fractional derivatives to seismic analysis of
base-isolated models, Earthquake engineering & structural dynamics, 19(2), 229–241,
(1990).
[7] K. Assaleh and W. M. Ahmad, Modeling of speech signals using fractional calculus,
2007 9th International Symposium on Signal Processing and Its Applications IEEE,
1–4 , (2007).
[8] F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to
mathematical models, World Scientific, (2022).
[9] R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus
to viscoelasticity, Journal of rheology, 27(3), 201–210, (1983).
[10] F. C. Meral, T. J. Royston and R. Magin, Fractional calculus in viscoelasticity: an experimental study, Communications in nonlinear science and numerical simulation, 15(4),
939–945, (2010).
[11] R. Koeller. Applications of fractional calculus to the theory of viscoelasticity, 299–307,
(1984).
[12] Z. E. A. Fellah, C. Depollier and M. Fellah. Application of fractional calculus to the
sound waves propagation in rigid porous materials: validation via ultrasonic measurements, Acta Acustica united with Acustica, 88(1), 34–39, (2002).
[13] B. Mathieu, P. Melchior, A. Oustaloup and C. Ceyral, Fractional differentiation for edge
detection, Signal Processing, 83(11), 2421–2432, (2003).
[14] V. V. Kulish and J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids
Eng., 124(3), 803–806, (2002).
[15] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives,
fractional differential equations, to methods of their solution and some of their applications, elsevier, (1998).
[16] R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional
derivative, Journal of computational and applied mathematics, 264, 65–70, (2014).
[17] T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the
frame of conformable derivatives, Advances in Difference Equations, 1–10, (2017).
[18] T. Abdeljawad, M. A. Horani and R. Khalil, Conformable fractional semigroups of operatorsn, J. Semigroup Theory Appl., (2015).
[19] M. Al-Refai and T. Abdeljawad. Fundamental results of conformable Sturm-Liouville
eigenvalue problems, Complexity, (2017).
[20] T. Abdeljawad, R. P. Agarwal, J. Alzabut, F. Jarad and A. Özbekler, Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives, Journal of Inequalities and Applications, 1–17, (2018).
[21] T. Abdeljawad, On conformable fractional calculus, Journal of computational and Applied Mathematics, 279, 57–66, (2015).
[22] P. J. Davis and P. Rabinowitz, Methods of numerical integration, Courier Corporation,
(2007).
[23] T. Allahviranloo, N. Ahmady and E. Ahmady, Numerical solution of fuzzy differential
equations by predictor–corrector method, Information sciences, 177(7), 1633–1647,
(2007).
[24] R. Klopfenstein and R. Millman, Numerical stability of a one-evaluation predictorcorrector algorithm for numerical solution of ordinary differential equations, Mathematics of Computation, 22(103), 557–564, (1968).
[25] E. Hairer, G. Wanner and O. Solving, H: Stiff and differential-algebraic problems, Berlin
[etc.]: Springer, (1991).
[26] K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3–22, (2002).
[27] K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of
Mathematical Analysis and Applications, 265(2), 229–248, (2002).
[28] K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional adams
method, Numerical algorithms, 36, 31–52, (2004).
[29] W. Zhong and L. Wang, Basic theory of initial value problems of conformable fractional
differential equations, Advances in Difference Equations, 2018, 1–14, (2018).
Published
2025-09-22
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