Lei Wang

College of Science, Binzhou University, Binzhou 256603, Shandong, China


 

Received: May 12, 2022
Accepted: September 10, 2022
Publication Date: November 24, 2022

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.


Download Citation: ||https://doi.org/10.6180/jase.202309_26(9).0005  


ABSTRACT


This paper aims to a present numerical method for solving a class of linear variable-order fractional boundary value problems. In this equation, some terms with fractional-order and some other ones of the correct degree appear in the equation. In order to handle this equation, we use the idea of the least-squares approximation method and Legendre polynomials. These polynomials constitute a system of the complete and orthogonal set. In recent years, useful mathematical properties have been extensively studied and used in various applications. Moreover, convergence analysis for the technique has also been studied in this manuscript. To demonstrate the validity and applicability of the technique, several numerical examples are provided in this article. Examining the results, we find that the approximations obtained in this paper are very accurate for the problem, and efficiently provide the approximation. The technique used in this article can be easily applied to solve similar problems. 


Keywords: FBVPs; LPs; Least-squares approximation approach; Residual error function; Approximate solutions


REFERENCES


  1. [1] M. A. Abdelkawy, A. Amin, and A. M. Lopes, (2022) “Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations" Computational and Applied Mathematics 41(1): 1–21.
  2. [2] Y.-M. Chen, Y.-Q. Wei, D.-Y. Liu, and H. Yu, (2015) “Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets" Applied Mathematics Letters 46: 83–88.
  3. [3] H. Rezazadeh, D. Kumar, T. A. Sulaiman, and H. Bulut, (2019) “New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation" Modern Physics Letters B 33(17): 1950196.
  4. [4] M. A. Abdelkawy, A. Z. Amin, A. M. Lopes, I. Hashim, and M. M. Babatin, (2021) “Shifted fractionalorder Jacobi collocation method for solving variable-order fractional integro-differential equation with weakly singular kernel" Fractal and Fractional 6(1): 19.
  5. [5] A. Dabiri, B. P. Moghaddam, and J. T. Machado, (2018) “Optimal variable-order fractional PID controllers for dynamical systems" Journal of Computational and Applied Mathematics 339: 40–48.
  6. [6] M. A. Abdelkawy, E. E. Mahmoud, K. M. Abualnaja, A.-H. Abdel-Aty, and S. Kumar, (2021) “Accurate spectral algorithm for two-dimensional variable-order fractional percolation equations" Mathematical Methods in the Applied Sciences 44(7): 6228–6238.
  7. [7] B. Ghanbari, (2021) “On novel nondifferentiable exact solutions to local fractional Gardner’s equation using an effective technique" Mathematical Methods in the Applied Sciences 44(6): 4673–4685.
  8. [8] I. Podlubnv, (1999) “Fractional differential equations academic press" San Diego, Boston 6:
  9. [9] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and applications of fractional differential equations. 204.elsevier, 2006.
  10. [10] C. F. Coimbra, (2003) “Mechanics with variable-order differential operators" Annalen der Physik 12(11-12): 692–703.
  11. [11] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2015) “Exact solutions for the fractional differential equations by using the first integral method" Nonlinear engineering 4(1): 15–22.
  12. [12] M. Eslami and H. Rezazadeh, (2016) “The first integral method for Wu–Zhang system with conformable timefractional derivative" Calcolo 53(3): 475–485.
  13. [13] H. Aminikhah, A. H. R. Sheikhani, and H. Rezazadeh, (2016) “Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method" Boletim da sociedade paranaense de matemática 34(2): 213–229.
  14. [14] A. Bhrawy and M. Zaky, (2016) “Numerical algorithm for the variable-order Caputo fractional functional differential equation" Nonlinear Dynamics 85(3): 1815–1823.
  15. [15] S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh, (2012) “Numerical techniques for the variable order time fractional diffusion equation" Applied Mathematics and Computation 218(22): 10861–10870.
  16. [16] L.-j. Xie, C.-l. Zhou, and S. Xu, (2018) “An effective computational method for solving linear multi-point boundary value problems" Applied Mathematics and Computation 321: 255–266.
  17. [17] C. Bota and B. C˘aruntu, (2012) “ε-Approximate polynomial solutions for the multi-pantograph equation with variable coefficients" Applied Mathematics and Computation 219(4): 1785–1792.
  18. [18] B. C˘aruntu and C. Bota, (2012) “Approximate polynomial solutions for nonlinear heat transfer problems using the squared remainder minimization method" International communications in heat and mass transfer 39(9): 1336-1341.
  19. [19] B. C˘aruntu and C. Bota, (2013) “Approximate polynomial solutions of the nonlinear Lane–Emden type equations arising in astrophysics using the squared remainder minimization method" Computer Physics Communications 184(7): 1643–1648.
  20. [20] D. S. Mohammed, (2014) “Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomial" Mathematical Problems in Engineering 2014:
  21. [21] Q. Wang, K. Wang, and S. Chen, (2014) “Least squares approximation method for the solution of Volterra– Fredholm integral equations" Journal of Computational and Applied Mathematics 272: 141–147.
  22. [22] J. Cao and Y. Qiu, (2016) “A high order numerical scheme for variable order fractional ordinary differential equation" Applied Mathematics Letters 61: 88–94.


    
 

0.9
2021CiteScore
 
 
42nd percentile
Powered by  Scopus

SCImago Journal & Country Rank

Enter your name and email below to receive latest published articles in Journal of Applied Science and Engineering.