Computational Algorithm for Approximating Fractional Derivatives of Functions
This paper presents an algorithmic approach for numerically solving Caputo fractional diﬀerentiation. The trapezoidal rule was modiﬁed, the new modiﬁcation was used to derive an algorithm to approximate fractional derivatives of order α > 0, the fractional derivative used was based on Caputo deﬁnition for a given function by a weighted sum of function and its ordinary derivatives values at speciﬁed points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerical approximation of Caputo fractional derivative for evaluating functions of fractional order. The study was conducted through some illustrative examples and analysis of error.
Keywords:Fractional Calculus, Finite difference Scheme, Modiﬁed trapezoidal rule
U. N. Katugampola, “A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES (COMMUNICATED BY CLAUDIO R. HENR´IQUEZHENR´HENR´IQUEZ),” Bulletin of Mathematical Analysis and Applications, vol. 6, pp. 1–15, 2014.
K. Diethelm et al., “Trends, directions for further research, and some open problems of fractional calculus,” Nonlinear Dynamics 2022 107:4, vol. 107, no. 4, pp. 3245–3270, Jan. 2022, doi:10.1007/S11071-021-07158-9.
G. E. Karniadakis, Handbook of Fractional Calculus with Applications, vol. 3. De Gruyter, 2019. doi:10.1515/9783110571684
J. E. Nápoles Valdes et al., “Analytical and Numerical Monotonicity Analyses for Discrete Delta Fractional Operators,” Mathematics 2022, Vol. 10, Page 1753, vol. 10, no. 10, p. 1753, May 2022, doi: 10.3390/MATH10101753.
C. F. Gerald, P. O. Wheatley, and Rahul Print O Pack), Applied numerical analysis. Pearson Educations Inc., 2004. Accessed: Dec. 29, 2022. [Online]. Available: https://books.google.com/books/about/Applied_Numerical_Analysis.html?id=8NJ_NlMCp9sC
J. H. Mathews and K. D. Fink, Numerical methods using MATLAB., 3rd ed. Prentice Hall, 1999.
V. E. Tarasov, Handbook of Fractional Calculus with Applications Applications in physics, part A, vol. 4. De Gruyter, 2019. doi:10.1515/9783110571707
V. E. Tarasov, Handbook of Fractional Calculus with Applications Applications in physics, part B, vol. 5. De Gruyter, 2019. doi:10.1515/9783110571721
V. E. Tarasov and V. v. Tarasova, Economic Dynamics with Memory Fractional Calculus Approach, vol. 8. De Gruyter, 2021. doi:10.1515/9783110627459
A. Kochubei and Y. Luchko, Handbook of Fractional Calculus with Applications Fractional differential equations. De Gruyter, 2019. doi: 10.1515/9783110571660
N. Abeye, M. Ayalew, D. L. Suthar, S. D. Purohit, and K. Jangid, “Numerical solution of unsteady state fractional advection–dispersion equation,” Arab Journal of Basic and Applied Sciences , vol. 29, no. 1, pp. 77–85, 2022, doi: 10.1080/25765299.2022.2064076.
Y. Luchko, “General Fractional Integrals and Derivatives with the Sonine Kernels,” Mathematics 2021, Vol. 9, Page 594, vol. 9, no. 6, p. 594, Mar. 2021, doi: 10.3390/MATH9060594.
Y. Luchko, “Operational calculus for the general fractional derivative and its applications,” Fract Calc Appl Anal, vol. 24, no. 2, pp. 338–375, Apr. 2021, doi: 10.1515/FCA-2021-0016
Y. Luchko and F. Martínez González, “General Fractional Integrals and Derivatives of Arbitrary Order,” Symmetry 2021, Vol. 13, Page 755, vol. 13, no. 5, p. 755, Apr. 2021, doi: 10.3390/SYM13050755.
V. E. Tarasov and W. S. Oates, “General Fractional Calculus: Multi-Kernel Approach,” Mathematics 2021, Vol. 9, Page 1501, vol. 9, no. 13, p. 1501, Jun. 2021, doi: 10.3390/MATH9131501.
U. Khristenko and B. Wohlmuth, “Solving time-fractional differential equation via rational approximation,” IMA Journal of Numerical Analysis, Feb. 2021, doi: 10.48550/arxiv.2102.05139.
L. Bennasr, “Sonine-Dimovski transform and spectral synthesis associated with the hyper-Bessel operator on the complex plane,” Fract Calc Appl Anal, vol. 25, no. 5, pp. 1852–1872, Oct. 2022, doi: 10.1007/S13540-022-00090-8
A. Kochubei, Y. Luchko, and J. A. T. Machado, Handbook of Fractional Calculus with Applications Basic theory, vol. 1. De Gruyter, 2019. doi: 10.1515/9783110571622
V. E. Tarasov, “On History of Mathematical Economics: Application of Fractional Calculus,” Mathematics 2019, Vol. 7, Page 509, vol. 7, no. 6, p. 509, Jun. 2019, doi: 10.3390/MATH7060509.
M. Moumen Bekkouche, I. Mansouri, and A. A. A. Ahmed, “Numerical solution of fractional boundary value problem with caputo-fabrizio and its fractional integral,” J Appl Math Comput, vol. 68, no. 6, pp. 4305–4316, Dec. 2022, doi: 10.1007/S12190-022-01708-Z
D. Yoon and D. You, “An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative,” Fract Calc Appl Anal, vol. 24, no. 5, pp. 1356–1379, Oct. 2021, doi: 10.1515/FCA-2021-0058
V. E. Tarasov and M. Lopes, “General Fractional Dynamics,” Mathematics 2021, Vol. 9, Page 1464, vol. 9, no. 13, p. 1464, Jun. 2021, doi: 10.3390/MATH9131464.
M. Huang et al., “Numerical solution of stochastic and fractional competition model in Caputo derivative using Newton method,” AIMS Mathematics 2022 5:8933, vol. 7, no. 5, pp. 8933–8952, 2022, doi: 10.3934/MATH.2022498.
K. Diethelm, R. Garrappa, A. Giusti, and M. Stynes, “Why fractional derivatives with nonsingular kernels should not be used,” Fract Calc Appl Anal, vol. 23, no. 3, pp. 610–634, Jun. 2020, doi: 10.1515/FCA-2020-0032
A. Traore and N. Sene, “Model of economic growth in the context of fractional derivative,” Alexandria Engineering Journal, vol. 59, no. 6, pp. 4843–4850, Dec. 2020, doi: 10.1016/J.AEJ.2020.08.047.
V. E. Tarasov, “General Fractional Vector Calculus,” Mathematics 2021, Vol. 9, Page 2816, vol. 9, no. 21, p. 2816, Nov. 2021, doi:10.3390/MATH9212816.
V. E. Tarasov, “Fractional dynamics with non-local scaling,” Commun Nonlinear Sci Numer Simul, vol. 102, p. 105947, Nov. 2021, doi: 10.1016/J.CNSNS.2021.105947.
R. Ashurov and Y. Fayziev, “On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations,” Fractal and Fractional 2022, Vol. 6, Page 41, vol. 6, no. 1, p. 41, Jan. 2022, doi: 10.3390/FRACTALFRACT6010041.
V. E. Tarasov, “General Non-Markovian Quantum Dynamics,” Entropy 2021, Vol. 23, Page 1006, vol. 23, no. 8, p. 1006, Jul. 2021, doi: 10.3390/E23081006.
M. Fečkan, M. Pospíšil, M. F. Danca, and J. R. Wang, “Caputo delta weakly fractional difference equations,” Fract Calc Appl Anal, vol. 25, no. 6, pp. 2222–2240, Dec. 2022, doi: 10.1007/S13540-022-00093-5
J. W. He and Y. Zhou, “Cauchy problem for non-autonomous fractional evolution equations,” Fract Calc Appl Anal, vol. 25, no. 6, pp. 2241–2274, Dec. 2022, doi: 10.1007/S13540-022-00094-4
P. Debnath, H. M. Srivastava, P. Kumam, and B. Hazarika, Fixed Point Theory and Fractional Calculus. Singapore: Springer Nature Singapore, 2022. doi: 10.1007/978-981-19-0668-8.
M. G. Ri and C. H. Yun, “Riemann-Liouville fractional derivatives of hidden variable recurrent fractal interpolation functions with function scaling factors and box dimension,” Chaos Solitons Fractals, vol. 156, p. 111793, Mar. 2022, doi:10.1016/J.CHAOS.2022.111793.
C. T. Ledesma, J. A. Rodríguez, and J. V. da C. Sousa, “Differential equations with fractional derivatives with fixed memory length,” Rendiconti del Circolo Matematico di Palermo, pp. 1–19, Jan. 2022, doi: 10.1007/S12215-021-00713-8
M. Edelman, “Cycles in asymptotically stable and chaotic fractional maps,” Nonlinear Dyn, vol. 104, no. 3, pp. 2829–2841, May 2021, doi: 10.1007/S11071-021-06379-2
L. Zhang, N. Zhang, and B. Zhou, “Solutions and stability for p-Laplacian differential problems with mixed type fractional derivatives,” International Journal of Nonlinear Sciences and Numerical Simulation, Jan. 2022, doi: 10.1515/IJNSNS-2021-0204
S. J. Achar, C. Baishya, P. Veeresha, and L. Akinyemi, “Dynamics of Fractional Model of Biological Pest Control in Tea Plants with Beddington–DeAngelis Functional Response,” Fractal and Fractional 2022, Vol. 6, Page 1, vol. 6, no. 1, p. 1, Dec. 2021, doi:10.3390/FRACTALFRACT6010001.
S. S. Ahmed, “Solving a System of Fractional-Order Volterra Integro-Differential Equations Based on the Explicit Finite Difference Approximation via the Trapezoid Method with Error Analysis,” Symmetry 2022, Vol. 14, Page 575, vol. 14, no. 3, p. 575, Mar. 2022, doi: 10.3390/SYM14030575.
A. F. Abdulhameed and Q. A. Memon, “An improved Trapezoidal rule for numerical integration,” J Phys Conf Ser, vol. 2090, no. 1, p. 012104, Nov. 2021, doi: 10.1088/1742-6596/2090/1/012104.
R. Garrappa, “Trapezoidal methods for fractional differential equations: Theoretical and computational aspects,” Math Comput Simul, vol. 110, no. 1, pp. 96–112, Apr. 2015, doi: 10.1016/J.MATCOM.2013.09.012.
D. Zhao, G. Gulshan, M. A. Ali, and K. Nonlaopon, “Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in q-Calculus,” Mathematics 2022, Vol. 10, Page 444, vol. 10, no. 3, p. 444, Jan. 2022, doi: 10.3390/MATH10030444.
J. O. Mamman and T. Aboiyar, “A Numerical Calculation of Arbitrary Integrals of Functions,” Advanced Journal of Graduate Research, vol. 7, no. 1, pp. 11–17, Oct. 2019, doi: 10.21467/ajgr.7.1.11-17.
Z. Odibat, “Approximations of fractional integrals and Caputo fractional derivatives,” Appl Math Comput, vol. 178, no. 2, pp. 527–533, Jul. 2006, doi: 10.1016/J.AMC.2005.11.072.
R. Gorenflo and F. Mainardi, “Fractional Calculus,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds. Springer, Vienna, 1997, pp. 223–276. doi: 10.1007/978-3-7091-2664-6_5.
A. Panda, J. Mohapatra, and N. R. Reddy, “A Comparative Study on the Numerical Solution for Singularly Perturbed Volterra Integro-Differential Equations,” Computational Mathematics and Modeling, vol. 32, no. 3, pp. 364–375, Jul. 2021, doi:10.1007/S10598-021-09536-9
How to Cite
Copyright (c) 2022 John Ojima Mamman
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Author(s) retains full copyright of their article and grants non-exclusive publishing right to this journal and its publisher "AIJR (India)". Author(s) can archive pre-print, post-print, and published version/PDF to any open access, institutional repository, social media, or personal website provided that Published source must be acknowledged with citation and link to publisher version.
Click here for more information on Copyright policy
Click here for more information on Licensing policy