Operational matrix for multi-order fractional differential equations with hermite polynomials

Hatice Yalman Koşunalp; Mustafa Gülsu

Operational matrix for multi-order fractional differential equations with hermite polynomials

Abstract

In this article, a new operational matrix of fractional integration of Hermite polynomials is derived to solve multi-order linear fractional differential equations (FDEs) with spectral tau approach. We firstly convert the FDEs into an integrated-form through multiple fractional integration in association with the Riemann-Liouville sense. This integral equation is then formulated as an algebraic equation system with Hermite polynomials. Finally, linear multi-order FDEs with initial conditions are solved with this method. We present exact and approximated solutions for a number of representative examples. Numerical results indicate that the proposed method provides a high degree of accuracy to solve the linear multi-order FDEs.

Keywords

References

[1] Carpinteri A, Mainardi F. Fractals and fractional calculus in continium mechanics. Vienna: Springer-Verlag Wien; 1997. [CrossRef]
[2] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. Toronto: John Wiley and Son;1993.
[3] Oldham KB, Spainer J. The fractional calculus, theory and applications of differentiation and integration to arbitrary order. Mineola, New York: Dover Publications; 2006.
[4] Podlubny I. Fractional differential equations. 1st ed. Cambridge, Massachusetts: Academic Press; 1999.
[5] Das S. Functional fractional calculus for system identification and controls. Heidelberg, Berlin: Springer; 2008.
[6] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam, Netherlands: Elsevier; 2006.
[7] Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus models and numerical methods. Hackensack, New Jersey; World Scientific Publishing; 2012. [CrossRef]
[8] Plonka A. Recent developments in dispersive kinetics. Progr React Kinet Mech 2000;25:109–127. [CrossRef]

[9] Allegrini P, Buiatti M, Grinolini P, West BL. Fractional brownian motion as a nonstationary process: An alternative paradigm for DNA sequences. J West Phys Rev 1998;57:558–567. [CrossRef]
[10] Bisquert J. Fractional diffusion in the multiple-trapping regime and revision of the equivalence with the continuous time random walk. Phys Rev Lett 2003;91:010602.
[11] Bailie RT, King ML. Fractional differencing and long memory processes. J Econom 1996;73:1–3. [CrossRef]
[12] Outsaloup A. La Commande CRONE: Commande robuste d’ordre non nntiere. Paris, France: Hermes; 1991.
[13] Bhrawy AK, Abdelkawy MA. A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J Comput Phys 2015;294:462–483. [CrossRef]
[14] Guo BY, Zhang XY. A new generalized laguerre approximation and its applications. J Comput Appl Math 2005;181:342–363. [CrossRef]
[15] Mikhailenko BG. Spectral laguerre method for the approximate solution of time-dependent problems. Appl Math Lett 1999;12:105–110. [CrossRef]
[16] Bhrawy AH, Alghamdi MM, Taha MT. A new modified generalized laguerre operational matrix of fractional integration for solving fractional differential equations on the half line. 2012;2012:179. [CrossRef]
[17] Ding XL, Jiang YL. Waveform relaxation methods for fractional differential equations with the Caputo derivatives. J Comput Appl Math 2012;16:573–594. [CrossRef]
[18] Bhrawy AH, Tharwat MM, Yıldırım A. A new formula for fractional integrals of chebyshev polynomials: application for solving multi-term fractional differential equations. Appl Math Model 2013;37:4245–4252. [CrossRef]
[19] Bhrawy AH, Alofi AS. The operational matrix of fractional integration for shifted chebyshev polynomials. Appl Math Lett 2013;26:25–31. [CrossRef]
[20] Doha EH, Bhrawy AH, Ezz-Eldien SS. A new jacobi operational matrix: An application for solving fractional differential equations. Appl Math Model 2013;36:4931–4943. [CrossRef]
[21] Bhrawy AH, Alghamdi MA. A shifted jacobi-gauss-lobatto collocation method for solving nonlinear fractional langevin equation. Bound Value Probl 2012;2012:62. [CrossRef]
[22] Bhrawy AH, Tharwat MM, Alghamdi MA. A new operational matrix of fractional integration for shifted jacobi polynomials. Bull Malays Math Sci Soc 2014;37:983–995.
[23] Saadatmandi A, Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput Math Appl 2014;3:1326–1336. [CrossRef]
[24] Akrami MH, Atabakzadeh MH, Erjaee GH. The operational matrix of fractional integration for shifted legendre polynomials. Iran J Sci Technol 2013;37:439–444. [CrossRef]
[25] Belgacem R, Bokhari A, Amir A. Bernoulli operational matrix of fractional derivative for solution of fractional differential equations. Gen Lett Math 2018;5:32– 46. [CrossRef]
[26] Poularikas AD. The handbook of formulas and tables for signal processing. 1st ed. Boca Raton: CRC Press; 1999. [CrossRef]
[27] Kumar P, Agrawal OP. An approximate method for numerical solution of fractional differential equations. Signal Process 2006;86:2602–2610. [CrossRef]

Details

Primary Language

English

Subjects

Structural Biology

Journal Section

Research Article

Authors

Hatice Yalman Koşunalp ^*
0000-0001-6313-862X
Türkiye

Mustafa Gülsu
0000-0001-6139-0266
Türkiye

Publication Date

August 1, 2024

Submission Date

December 26, 2022

Acceptance Date

April 4, 2023

Published in Issue

Year 2024 Volume: 42 Number: 4

IZ

https://izlik.org/JA74EZ56SL

Cite

RIS / Bibtex

APA

Yalman Koşunalp, H., & Gülsu, M. (2024). Operational matrix for multi-order fractional differential equations with hermite polynomials. Sigma Journal of Engineering and Natural Sciences, 42(4), 1050-1057. https://izlik.org/JA74EZ56SL

AMA

1.Yalman Koşunalp H, Gülsu M. Operational matrix for multi-order fractional differential equations with hermite polynomials. SIGMA. 2024;42(4):1050-1057. https://izlik.org/JA74EZ56SL

Chicago

Yalman Koşunalp, Hatice, and Mustafa Gülsu. 2024. “Operational Matrix for Multi-Order Fractional Differential Equations With Hermite Polynomials”. Sigma Journal of Engineering and Natural Sciences 42 (4): 1050-57. https://izlik.org/JA74EZ56SL.

EndNote

Yalman Koşunalp H, Gülsu M (August 1, 2024) Operational matrix for multi-order fractional differential equations with hermite polynomials. Sigma Journal of Engineering and Natural Sciences 42 4 1050–1057.

IEEE

[1]H. Yalman Koşunalp and M. Gülsu, “Operational matrix for multi-order fractional differential equations with hermite polynomials”, SIGMA, vol. 42, no. 4, pp. 1050–1057, Aug. 2024, [Online]. Available: https://izlik.org/JA74EZ56SL

ISNAD

Yalman Koşunalp, Hatice - Gülsu, Mustafa. “Operational Matrix for Multi-Order Fractional Differential Equations With Hermite Polynomials”. Sigma Journal of Engineering and Natural Sciences 42/4 (August 1, 2024): 1050-1057. https://izlik.org/JA74EZ56SL.

JAMA

1.Yalman Koşunalp H, Gülsu M. Operational matrix for multi-order fractional differential equations with hermite polynomials. SIGMA. 2024;42:1050–1057.

MLA

Yalman Koşunalp, Hatice, and Mustafa Gülsu. “Operational Matrix for Multi-Order Fractional Differential Equations With Hermite Polynomials”. Sigma Journal of Engineering and Natural Sciences, vol. 42, no. 4, Aug. 2024, pp. 1050-7, https://izlik.org/JA74EZ56SL.

Vancouver

1.Hatice Yalman Koşunalp, Mustafa Gülsu. Operational matrix for multi-order fractional differential equations with hermite polynomials. SIGMA [Internet]. 2024 Aug. 1;42(4):1050-7. Available from: https://izlik.org/JA74EZ56SL