Modified 5-point fractional formula with Richardson extrapolation

Iqbal M. Batiha; Shameseddin Alshorm; Iqbal Jebril; Amjed Zraiqat; Zaid Momani; Shaher Momani; Iqbal M. Batiha; Shameseddin Alshorm; Iqbal Jebril; Amjed Zraiqat; Zaid Momani; Shaher Momani

doi:10.3934/math.2023480

AIMS Mathematics

2023, Volume 8, Issue 4: 9520-9534. doi: 10.3934/math.2023480

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Modified 5-point fractional formula with Richardson extrapolation

1.
Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan
2.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
3.
Department of Civil Engineering, The University of Texas at Arlington, Box 19308,428 Nedderman Hall, Arlington, TX 76019, USA
4.
Department of Mathematics, The University of Jordan, Amman 11942, Jordan

Received: 28 October 2022 Revised: 17 January 2023 Accepted: 13 February 2023 Published: 20 February 2023
MSC : 26A33, 34A08, 34K37

In this paper, we establish a novel fractional numerical modification of the 5-point classical central formula; called the modified 5-point fractional formula for approximating the first fractional-order derivative in the sense of the Caputo operator. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability.
- Richardson extrapolation,
- Riemann-Liouville fractional derivative and integral,
- Lagrange interpolating polynomial,
- Caputo derivative
Citation: Iqbal M. Batiha, Shameseddin Alshorm, Iqbal Jebril, Amjed Zraiqat, Zaid Momani, Shaher Momani. Modified 5-point fractional formula with Richardson extrapolation[J]. AIMS Mathematics, 2023, 8(4): 9520-9534. doi: 10.3934/math.2023480

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Abstract

In this paper, we establish a novel fractional numerical modification of the 5-point classical central formula; called the modified 5-point fractional formula for approximating the first fractional-order derivative in the sense of the Caputo operator. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability.

References

[1]	A. C. Aitken, On interpolation by iteration of proportional parts, without the use of differences, Proc. Edinb. Math. Soc., 3 (1932), 56–76. https://doi.org/10.1017/S0013091500013808 doi: 10.1017/S0013091500013808
[2]	W. F. Ames, Numerical methods for partial differential equations, Academic Press, 1992.
[3]	O. Axelsson, V. A. Barker, Finite element solution of boundary value problems: theory and computation, Academic Press, 1984.
[4]	R. B. Albadarneh, I. M. Batiha, M. Zurigat, Numerical solutions for linear fractional differential equations of order $1 < \alpha < 2$ using finite difference method (FFDM), Int. J. Math. Comput. Sci., 16 (2016), 103–111.
[5]	R. B. Albadarneh, I. M. Batiha, A. Adwai, N. Tahat, A. K. Alomari, Numerical approach of Riemann-Liouville fractional derivative operator, Int. J. Electr. Comput. Eng., 11 (2021), 5367–5378.
[6]	E. L. Allgower, K. Georg, Numerical continuation methods: an introduction, Berlin, Heidelberg: Springer-Verlag, 1990. https://doi.org/10.1007/978-3-642-61257-2
[7]	A. V. Aho, J. E. Hopcroft, The design and analysis of computer algorithms, Pearson Education India, 1974.
[8]	M. Arshad, D. C. Lu, J. Wang, (N+1)-dimensional fractional reduced differential transform method for fractional order partial differential equations, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 509–519. https://doi.org/10.1016/j.cnsns.2017.01.018 doi: 10.1016/j.cnsns.2017.01.018
[9]	U. M. Ascher, R. M. M. Mattheij, R. D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Englewood Cliffs: Prentice-Hall, 1988.
[10]	I. K. Argyros, F. Szidarovszky, The theory and applications of iteration methods, Boca Raton: CRC Press, 1993. https://doi.org/10.1201/9780203719169
[11]	I. Batiha, S. M. Alshorm, I. Jebril, M. A. Hammad, A brief review about fractional calculus, Int. J. Open Problems Complex Anal., 15 (2022), 39–56.
[12]	I. M. Batiha, S. Alshorm, A. Ouannas, S. Momani, O. Y. Ababneh, M. Albdareen, Modified three-point fractional formulas with Richardson extrapolation, Mathematics, 10 (2022), 1–16. https://doi.org/10.3390/math10193489 doi: 10.3390/math10193489
[13]	I. M. Batiha, R. El-Khazali, A. AlSaedi, S. Momani, The general solution of singular fractional-order linear time-invariant continuous systems with regular pencils, Entropy, 20 (2018), 1–14. https://doi.org/10.3390/e20060400 doi: 10.3390/e20060400
[14]	R. L. Burden, J. D. Faires, Numerical analysis, 9 Eds., Thomson Brooks/Cole, 2005.
[15]	R. El-Khazali, I. M. Batiha, S. Momani, Approximation of fractional-order operators, In: Fractional calculus, Singapore: Springer, 2018. https://doi.org/10.1007/978-981-15-0430-3_8
[16]	D. C. Joyce, Survey of extrapolation processes in numerical analysis, SIAM Review, 13 (1971), 435–490. https://doi.org/10.1137/1013092 doi: 10.1137/1013092
[17]	V. Martynyuk, M. Ortigueira, M. Fedula, O. Savenko, Methodology of electrochemical capacitor quality control with fractional order model, AEU-Int. J. Electr. Commun., 91 (2018), 118–124. https://doi.org/10.1016/j.aeue.2018.05.005 doi: 10.1016/j.aeue.2018.05.005
[18]	M. D. Ortigueira, J. T. Machado, Fractional derivatives: The perspective of system theory, Mathematics, 7 (2019), 1–14. https://doi.org/10.3390/math7020150 doi: 10.3390/math7020150
[19]	M. D. Ortigueira, A. G. Batista, On the relation between the fractional Brownian motion and the fractional derivatives, Phys. Lett. A, 372 (2008), 958–968. https://doi.org/10.1016/j.physleta.2007.08.062 doi: 10.1016/j.physleta.2007.08.062
[20]	V. De Santis, V. Martynyuk, A. Lampasi, M. Fedula, M. D. Ortigueira, Fractional-order circuit models of the human body impedance for compliance tests against contact currents, AEU-Int. J. Electr. Commun., 78 (2017), 238–244. https://doi.org/10.1016/j.aeue.2017.04.035 doi: 10.1016/j.aeue.2017.04.035
[21]	A. H. Salas, M. A. Hammad, B. M. Alotaibi, L. S. El-Sherif, S. A. El-Tantawy, Analytical and numerical approximations to some coupled forced damped duffing oscillators, Symmetry, 14 (2022), 1–13. https://doi.org/10.3390/sym14112286 doi: 10.3390/sym14112286
[22]	I. Talbi, A. Ouannas, A. Khennaoui, A. Berkane, I. M. Batiha, G. Grassi, et al., Different dimensional fractional-order discrete chaotic systems based on the Caputo $h$-difference discrete operator: dynamics, control, and synchronization, Adv. Differ. Equ., 2020 (2020), 1–15. https://doi.org/10.1186/s13662-020-03086-x doi: 10.1186/s13662-020-03086-x
[23]	Q. Wang, Homotopy perturbation method for fractional KdV equation, Appl. Math. Comput., 190 (2007), 1795–1802. https://doi.org/10.1016/j.amc.2007.02.065 doi: 10.1016/j.amc.2007.02.065
[24]	Z. Zlatev, I. Dimov, I. Farago, A. Havasiy, Richardson extrapolation, Berlin, Boston: De Gruyter, 2018. https://doi.org/10.1515/9783110533002

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