THERMAL SCIENCE

International Scientific Journal

Sonali M. Narsale

,

Hossein Jafari

,

Ram Kishun Lodhi

A NUMERICAL STUDY FOR SOLVING MULTI-TERM FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS

ABSTRACT
In this article, we extended operational matrices using orthonormal Boubaker polynomials of Riemann-Liouville fractional integration and Caputo derivative to find numerical solution of multi-term fractional-order differential equations (FDE). The proposed method is utilized to convert FDE into a system of algebraic equations. The convergence of the method is proved. Examples are given to explain the simplicity, computational time and accuracy of the method.
KEYWORDS
fractional differential equations, orthonormal Boubaker polynomials, Gram-Schmidt process, operational matrix, covergence analysis
PAPER SUBMITTED: 2022-02-01
PAPER REVISED: 2022-03-04
PAPER ACCEPTED: 2022-03-14
PUBLISHED ONLINE: 2023-04-08
DOI REFERENCE: https://doi.org/10.2298/TSCI23S1401N
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE Special issue 1, PAGES [401 - 410]
REFERENCES
  1. Ross, B., The Development of Fractional Calculus 1695-1900, Historia Mathematica, 1 (1977), 4, pp. 75-89
  2. Hristov, J., BioHeat Models Revisited: Concepts, Derivations, Non-Dimensalization and Fractionalization Approaches, Front. Phys., 7 (2019), 189, pp. 1-36
  3. Bolandtalat, A., et al., Numerical Solutions of Multi-Order Fractional Differential Equations by Boubaker Polynomials, Open Physics, 1 (2016), 14, pp. 226-230
  4. Bhrawy, A. H., et al., A Review of Operational Matrices and Spectral Techniques for Fractional Calculus, Non-Linear Dynamics, 3 (2015), 81, pp. 1023-1052
  5. Yang, Y., et al., Legendre Polynomials Operational Matrix Method for Solving Fractional Partial Differential Equations with Variable Coefficients, Mathematical Problems in Engineering, 2015 (2015), ID915195
  6. Tural-Polat, S. N., Turan Dincel, A., Numerical Solution Method for Multi-Term Variable Order Fractional Differential Equations by Shifted Chebyshev Polynomials of the Third Kind, Alexandria Engineering Journal, 7 (2022), 61, pp. 5145-5153
  7. Ganji, R. M., Jafari, H., A Numerical Approach for Multi-Variable Orders Differential Equations Using Jacobi Polynomials, International Journal of Applied and Computational Mathematics, 5 (2019), Feb., 34
  8. El-Sayed, A. A., et al., A Novel Jacobi Operational Matrix for Numerical Solution of Multi-Term Variable-Order Fractional Differential Equations, Journal of Taibah University for Science, 1 (2020), 14, pp. 963-974
  9. Mayada, M. A., A New Operational Matrix of Derivative for Orthonormal Bernstein Polynomial's, Baghdad Science Journal, 3 (2014), 11, pp. 1295-1300
  10. Youssri, H., Y., A New Operational Matrix of Caputo Fractional Derivatives of Fermat Polynomials: An Application for Solving the Bagley-Torvik Equation, Advances in Difference Equations, 1 (2017), Mar., 73
  11. Xinxiu, L., Operational Method for Solving Fractional Differential Equations Using Cubic B-Spline Aproximation, International Journal of Computer Mathematics, 12 (2014), 91, pp. 2584-2602
  12. Lin, S., et al., Shifted Legendre Polynomials Algorithm Used for the Numerical Analysis of Viscoelastic Plate with a Fractional Order Model, Mathematics and Computers in Simulation, 193 (2022), Mar., pp. 190-203
  13. Shazia, S., Mujeeb ur Rehman, Solution of Fractional Boundary Value Problems by ψ-Shifted Operational Matrices, AIMS Mathematics, 4 (2022), 7, pp. 6669-6693
  14. Nazeer, A. K., On Orthogonalization of Boubaker Polynomials, Journal of Mechanics of Continua and Mathematical Sciences, 11 (2020), 15, pp. 119-131
  15. Kobra, R., et al., The Boubaker Polynomials and Their Application Solve Fractional Optimal Control Problems, Non-Linear Dynamics, 2 (2017), 88, pp. 1013-1026
  16. Abdelkrim, B., et al., A New Operational Matrix Based on Boubaker Polynomials for Solving Fractional Emden-Fowler Problem, On-line first, doi.org/10.48550/arXiv.2022.09787, 2022
  17. Tinggang, Z., Yongjun, L., Boubaker Polynomial Spectral-like Method for Numerical Solution of Differential Equations, Proceedings, 2nd Int. Con. on Electronic, Network, and Computer Eng., Yinchuan, China, 2016, pp. 276-281
  18. Podlubny, I., Fractional Differential Equations, Academic Press, London, UK, 1999
  19. Kashem, B., et al., Some Results for Orthonormal Boubaker Polynomials with Application, Journal of Southwest Jiaotong University, 3 (2020), 55
  20. Maw, L.W., et al., Generalization of Generalized Orthogonal Polynomial Operational Matrices for Fractional and Operational Calculus, International Journal of Systems Science, 18 (1987), pp. 931-943
PDF VERSION [DOWNLOAD]
A numerical study for solving multi-term fractional-order differential equations

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence