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.
Downloads
References
P. L. Butzer, U. Westphal, An introduction to fractional calculus. Applications of Fractional Calculus in Physics, 1-85, (2000).
J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation 16(3), 1140-1153, (2011).
A. S. Shaikh, I. N. Shaikh, 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).
R. L. Magin, Fractional calculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications 59(5), 1586-1593, (2010).
M. Dalir, M. Bashour, Applications of fractional calculus. Applied Mathematical Sciences 4(21), 1021-1032, (2010).
C. G. Koh, J. M. Kelly, Application of fractional derivatives to seismic analysis of base-isolated models. Earthquake Engineering and Structural Dynamics 19(2), 229-241, (1990).
K. Assaleh, W. M. Ahmad, Modeling of speech signals using fractional calculus. 9th International Symposium on Signal Processing and Its Applications IEEE, 1-4, (2007).
A. K. Yadav, E. Carrera, M. Marin, M. I. A. Othman, Reflection of hygrothermal waves in a Nonlocal Theory of coupled thermo-elasticity. Mech. Adv. Mater. Struct. 31(5), 1083-1096, (2024).
M. Marin, I. Abbas, and R. Kumar, Relaxed Saint-Venant principle for thermoelastic micropolar diffusion. Struct. Eng. Mech. 51(4), 651-662, (2014).
M. M. Bhatti, M. Marin, R. Ellahi, and I. M. Fudulu, Insight into the dynamics of EMHD hybrid nanofluid (ZnO/CuO-SA) flow through a pipe for geothermal energy applications. J. Therm. Anal. Calorim. 148(24), 14261-14273, (2023).
F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific, (2022).
R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology 27(3), 201-210, (1983).
F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity: an experimental study. Communications in Nonlinear Science and Numerical Simulation 15(4), 939-945, (2010).
R. Koeller, Applications of fractional calculus to the theory of viscoelasticity. 299-307, (1984).
Z. E. A. Fellah, C. Depollier, 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).
B. Mathieu, P. Melchior, A. Oustaloup, C. Ceyral, Fractional differentiation for edge detection. Signal Processing 83(11), 2421-2432, (2003).
V. V. Kulish, J. L. Lage, Application of fractional calculus to fluid mechanics. Journal of Fluids Engineering 124(3), 803-806, (2002).
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).
R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. Journal of Computational and Applied Mathematics 264, 65-70, (2014).
T. Abdeljawad, J. Alzabut, F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives. Advances in Difference Equations, 1-10, (2017).
T. Abdeljawad, M. A. Horani, R. Khalil, Conformable fractional semigroups of operators. Journal of Semigroup Theory and Applications, (2015).
M. Al-Refai, T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems. Complexity, (2017).
T. Abdeljawad, R. P. Agarwal, J. Alzabut, F. Jarad, A. Özbekler, Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives. Journal of Inequalities and Applications, 1-17, (2018).
T. Abdeljawad, On conformable fractional calculus. Journal of Computational and Applied Mathematics 279, 57-66, (2015).
T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictor-corrector method. Information Sciences 177(7), 1633-1647, (2007).
R. Klopfenstein, R. Millman, Numerical stability of a one-evaluation predictor-corrector algorithm for numerical solution of ordinary differential equations. Mathematics of Computation 22(103), 557-564, (1968).
E. Hairer, G. Wanner, O. Solving, H: Stiff and differential-algebraic problems. Springer, Berlin, (1991).
K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29, 3-22, (2002).
C. J. Zúñiga-Aguilar, H. M. Romero-Ugalde, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks. Chaos, Solitons & Fractals 103, 382-403, (2017).
Y. T. Toh, C. Phang, J. R. Loh, New predictor-corrector scheme for solving nonlinear differential equations with Caputo-Fabrizio operator. Mathematical Methods in the Applied Sciences 42(1), 175-185, (2019).
C. W. H. Green, Y. Liu, Y. Yan, Numerical methods for Caputo-Hadamard fractional differential equations with graded and non-uniform meshes. Mathematics 9(21), 2728, (2021).
P. J. Davis, P. Rabinowitz, Methods of numerical integration. Courier Corporation, (2007).
K. Diethelm, N. J. Ford, Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265(2), 229-248, (2002).
K. Diethelm, N. J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31-52, (2004).
W. Zhong and L. Wang, Basic theory of initial value problems of conformable fractional differential equations. Adv. Differ. Equ. 2018, 1-14, (2018).
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
