The Limited Validity of the Fractional Euler Finite Difference Method and an Alternative Definition of the Caputo Fractional Derivative to Justify Modification of the Method

6e4d1b86-6376-4f4e-b085-7a254c36fdda20231109091035562wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051919202291920222210.37394/23206.2023.22https://wseas.com/journals/mathematics/2023.phpThe Limited Validity of the Fractional Euler Finite Difference Method and an Alternative Definition of the Caputo Fractional Derivative to Justify Modification of the MethodDominic P.Clemence-MkhopeDepartment of Mathematics and Statistics, North Carolina A&T State University, 1601 East Market Street, Greensboro, NC 27411, USAZacharyDentonDepartment of Mathematics and Statistics, North Carolina A&T State University, 1601 East Market Street, Greensboro, NC 27411, USAA method, advanced as the fractional Euler finite difference method (FEFDM), a general method for the finite difference discretization of fractional initial value problems (IVPs) for 0<α≤1 for the Caputo derivative, is shown to be valid only for α=1. This is accomplished by establishing, through a recently proposed generalized difference quotient representation of the fractional derivative, that the FEFDM is valid only if a property of the Mittag-Leffler function holds that has only been shown to be valid only for α=1. It is also shown that the FEFDM is inconsistent with the exact discretization of the IVP for the Caputo fractional relaxation equation. The generalized derivative representation is also used to derive a modified generalized Euler’s method, its nonstandard finite difference alternative, their improved Euler versions, and to recover a recent result by Mainardi relating the Caputo and conformable derivatives.1192023119202383184191https://wseas.com/journals/mathematics/2023/b845106-047(2023).pdf10.37394/23206.2023.22.91https://wseas.com/journals/mathematics/2023/b845106-047(2023).pdfI. Podlubny, Fractional Differential Equations, Academic Press, 1999. M. Dalir and M. Bashour, Applications of Fractional Calculus Applied Mathematical Sciences, 4(21), 2010, pp.1021–1032 10.1111/j.1365-246x.1967.tb02303.xM. Caputo, Linear Models of Dissipation whose Q is almost Frequency Independent-II. Geophys. J. R. ustr. SOC., 13, 1967, pp.529– 539 10.1016/j.cma.2004.06.006K. Diethelm, N.J. Ford, A.D. Freed, and Yu. Luchko, Algorithms for the fractional calculus: a selection of numerical methods. Comput. Meth. Appl. Mech. Engrg., 194, 2005, pp.743–773. Z. M.Odibat and S. Momani, An Algorithm for the numerical solution of differential equations of fractional order, Journal of applied mathematics & informatics, 26 (12), 2008, pp.15–27 10.1142/s0218127412300145C. Li and F. Zeng, Finite Difference Methods For Fractional Differential Equations. International Journal of Bifurcation and Chaos, 22(4), 2012, 1230014 (28 pages) 10.1142/8180D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus: Models and numerical methods, World Scientific, 2012. 10.1016/j.jcp.2013.11.017G-h.Gao, Z-z. Sun, and H-w. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, Journal of Computational Physics, 259, 2014, pp.33–50 10.3390/math6020016R. Garrappa, Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. Mathematics. 6(2), 2018:16. https://doi.org/10.3390/math6020016 10.20454/jmmnm.2015.959M.M. Khadera and M. Adel, Numerical treatment of the fractional SIRC model and influenza A using generalized Euler method, J. of Modern Methods in Numerical Math 6:1, 2015, pp.44–56 D. Yaro, S.K. Omari-Sasu, P. Harvim, A.W. Saviour, and B.A. Obeng, Generalized Euler Method for Modeling Measles with Fractional Differential Equations, International Journal of Innovative Research & Development, 4(4), 2015, pp.358–366 D. Vivek, K. Kanagarajan, and S. Harikrishnan, Numerical solution of fractional-order logistic equations by fractional Euler’s method, International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2016, 4(III), pp.775–780 10.1186/s13662-019-2055-yY.Y.Y. Noupoue, Y. Tandoğdu, and M. Awadalla, On numerical techniques for solving the fractional logistic differential equation, Advances in Difference Equations, 108 (2019), 2019, pp.1–13. https://doi.org/10.1186/s13662-019-2055-y 10.1080/01630563.2012.706673C. Li and F. Zeng, The Finite Difference Methods for Fractional Ordinary Differential Equations, Numerical Functional Analysis and Optimization, 34(2), 2013, pp.149–179, DOI: 10.1080/01630563.2012.706673: https://doi.org/10.1080/01630563.2012.70667 3 10.1515/jacsm-2015-0010N.A. Khan, O.A. Razzaq, and F. Riaz, Numerical Simulations For Solving Fuzzy Fractional Differential Equations By MaxMin Improved Euler Methods, JACSM, 7(1), 2015, pp.53–83 10.22401/anjs.00.1.19F.S. Fadhel and H.H. Khayoon, Numerical Solution of Ordinary Differential Equations of Fractional Order Using Variable Step Size Method, Special Issue: 1st Scientific International Conference, College of Science, Al-Nahrain University, 21-22/11/2017, Part I, pp.143–149, [Online], https://www.iasj.net/iasj/download/875d174d 6dd06236 (Accessed Date: October 17, 2023) 10.21608/joems.2018.9460H.F. Ahmed, Fractional Euler Method; An Effective Tool For Solving Fractional Differential Equations, Journal of the Egyptian Mathematical Society, 26(1), 2018, pp.38–43. DOI: 10.21608/JOEMS.2018.9460 M.S. Mechee and S.H. Aidib, Generalized Euler and Runge-Kutta methods for solving classes of fractional ordinary differential equations, Int. J. Nonlinear Anal. Appl. 13(1), 2022, pp.1737–1745 http://dx.doi.org/10.22075/ijnaa.2022.5788 10.1016/j.amc.2006.07.102Z. Odibat and N. Shawagfeh, Generalized Taylor’s formula. Mathematics and Computation, 186(1), 2007, 286–293. DOI: 10.1016/j.amc.2006.07.102 10.3390/mca27010004D.P. Clemence-Mkhope, Spectral Non-integer Derivative Representations and the Exact Spectral Derivative Discretization Finite Difference Method for the Fokker-Planck Equation. 2021, [Online], https://ui.adsabs.harvard.edu/link_ga teway/2021arXiv210602586C/arxiv:2106.025 86 (Accessed Date: October 17, 2023) R.E. Mickens, Nonstandard Finite Difference Schemes: Methodology And Applications, World Scientific Publishing Company, 2020. 10.1016/j.cam.2014.01.002R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264, pp.65–70. 10.3390/mca26040066D.P. Clemence-Mkhope and B.G.B. Clemence-Mkhope, The Limited Validity of the Conformable Euler Finite Difference Method and an Alternate Definition of the Conformable Fractional Derivative to Justify Modification of the Method. Mathematical and Computational Applications. 2021; 26(4):66. https://doi.org/10.3390/mca26040066 10.1016/j.jmaa.2010.04.031J. Peng and K. Li, A note on property of the Mittag-Leffler function. Journal of Mathematical Analysis and Applications, 370(2), 2010, pp.635–638 10.3390/math6010008F. Mainardi, A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients. Reprinted from: Mathematics 2018, 6,8, DOI: 10.3390/math6010008 10.4236/ajcm.2016.62015S.S. Zeid, M. Yousefi, and A.V. Kamyad, Approximate Solutions for a Class of Fractional-Order Model of HIV Infection via Linear Programming Problem. American Journal of Computational Mathematics, 6, 2016, pp.141–152. DOI: 10.4236/ajcm.2016.62015. 10.1016/j.mcm.2009.04.019Y. Ding and H. Ye, A fractional-order differential equation model of HIV infection of CD4+ T-cells, Mathematical and Computer Modelling, 50 (3–4), 2009, pp.386–392, https://doi.org/10.1016/j.mcm.2009.04.019.