Numerical Solution of Fractional Order Differential Equations by Chebyshev Least Squares Approximation Method

Ajimot F. Adebisi; Ohigweren A. Uwaheren; Wasiu A. Oseni; Christie Y. Ishola; Olumuyiwa James Peter

doi:10.37905/euler.v13i1.31034

Numerical Solution of Fractional Order Differential Equations by Chebyshev Least Squares Approximation Method

Ajimot F. Adebisi, Ohigweren A. Uwaheren, Wasiu A. Oseni, Christie Y. Ishola, Olumuyiwa James Peter

Abstract

In this paper, fractional differential equations (FDEs) are solved numerically using the least squares method (LSM). Shifted Chebyshev polynomials are used as the basis functions, and the results are compared with the exact solutions. Several numerical examples are presented to illustrate the theoretical results and are also compared with the outcomes obtained from other numerical methods. It is found that the results of the proposed approximate method converge rapidly to the exact solutions.

Keywords

Fractional Differential Equations; Least Squares Methods; Shifted Chebychev Polynomials

Full Text:

PDF

References

S. Ahmed dan T. Elzaki, "The solution of nonlinear Volterra integrodifferential equations of second kind by combine Sumudu transforms and Adomian decomposition method," International Journal of Advanced and Innovative Research, vol. 2, no. 12, pp. 90–93, 2014.

N. M. Dabwan dan Y. Q. Hasan, "Solution of Fractional Differential Equations by Modified Adomian Decomposition Method," Advances in Mathematics: Scientific Journal, vol. 9, no. 7, pp. 4761–4767, 2020. doi: 10.37418/amsj.9.7.42.

A. Khalouta dan A. Kadem, "A New Method to Solve Fractional Differential Equations: Inverse Shehu Transform Method," International Journal: Applications and Applied Mathematics, vol. 14, no. 2, pp. 926–941, 2019.

H. Bulut, H. M. Baskonus, dan F. B. M. Belgacem, "The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method," Abstract and Applied Analysis, vol. 2013, Article ID 203875, pp. 1–6, 2013. doi: 10.1155/2013/203875.

B. Ibis, M. Bayram, dan A. G. Agargun, "Applications of Fractional Differential Transform Method to Fractional Differential-Algebraic Equations," European Journal of Pure and Applied Mathematics, vol. 4, no. 2, pp. 129–141, Mei 2011. Tersedia secara daring: https://www.ejpam.com/index.php/ejpam/article/view/1172

A. Kehaili, A. Hakem, dan A. Benali, "Homotopy Perturbation Transform Method for Solving the Partial and the Time-Fractional Differential Equations with Variable Coefficients," Global Journal of Pure and Applied Sciences, vol. 26, no. 1, pp. 35–55, Juni 2020. doi: 10.4314/gjpas.v26i1.6.

M. S. Mohamed, M. R. Alharthi, dan R. A. Alotabi, "Solving Fractional Integro-Differential Equations by Homotopy Analysis Transform Method," International Journal of Pure and Applied Mathematics, vol. 106, no. 4, pp. 1037–1055, 2016. doi: 10.12732/ijpam.v106i4.6.

C. Bota, B. Caruntu, D. Tucu, M. Lapadat, dan M. S. Paşca, "A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order," Mathematics, vol. 8, no. 8, p. 1336, 2020. doi: 10.3390/math8081336.

T. Oyedepo, O. B. Akinduko, O. E. Faniyi, dan E. P. Okperhie, "Bernstein Least-Squares Technique for Solving Fractional Integro-Differential Equations," Science World Journal, vol. 14, no. 3, pp. 56–60, Sep. 2019.

O. A. Taiwo dan M. O. Fesojaye, "Perturbation Least-Squares Chebyshev Method for Solving Fractional Order Integro-Differential Equations," Theoretical Mathematics and Applications, vol. 5, no. 4, pp. 37–47, 2015.

O. A. Uwaheren, A. F. Adebisi, dan O. A. Taiwo, "Perturbed Collocation Method For Solving Singular Multi-order Fractional Differential Equations of Lane-Emden Type," Journal of the Nigerian Society of Physical Sciences, vol. 3, pp. 141–148, 2020. doi: 10.46481/jnsps.2020.69.

A. M. S. Mahdy dan R. T. Shwayye, "Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Laguerre Polynomials Pseudo-Spectral Method," International Journal of Scientific & Engineering Research, vol. 7, no. 4, pp. 1589–1596, Apr. 2016.

S. N. Shehab, H. A. Ali, dan H. M. Yaseen, "Least Squares Method for Solving Integral Equations with Multiple Time Lags," Engineering and Technology Journal, vol. 28, no. 10, pp. 1893–1899, 2010.

O. A. Uwaheren dan O. A. Taiwo, "A Collocation Technique Based on an Orthogonal Polynomial for Solving Multi-Fractional Order Integro-Differential Equations," Journal of the Mathematical Society of Nigeria (ABACUS), vol. 43, pp. 215–220, 2016.

A. Verma dan M. Kumar, "Numerical Solution of Lane–Emden Type Equations Using Multilayer Perceptron Neural Network Method," International Journal of Applied and Computational Mathematics, vol. 5, no. 5, p. 141, Okt. 2019. doi: 10.1007/s40819-019-0728-6.

Y. Öztürk, "An Efficient Numerical Algorithm for Solving System of Lane–Emden Type Equations," Nonlinear Engineering, vol. 8, no. 1, pp. 429–437, 2019. doi: 10.1515/nleng-2018-0062.

M. Weilbeer, "Efficient Numerical Methods for Fractional Differential Equations and Their Analytical Background," Disertasi Ph.D., Carl-Friedrich-Gauss Faculty of Mathematics and Computer Science, Technische Universität Braunschweig, 2005.

P. Guo, "The Adomian Decomposition Method for a Type of Fractional Differential Equations," Journal of Applied Mathematics and Physics, vol. 7, no. 10, pp. 2459–2466, Okt. 2019. doi: 10.4236/jamp.2019.710166.

W. Al-Hayani, L. Alzubaidy, dan A. Entesar, "Solutions of Singular IVP’s of Lane-Emden Type by Homotopy Analysis Method with Genetic Algorithm," Applied Mathematics and Information Sciences, vol. 11, no. 2, pp. 407–416, 2017. doi: 10.18576/amis/110208.

M. M. Khader dan K. M. Saad, "A numerical study using Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation," International Journal of Biomathematics, vol. 11, no. 8, p. 1850099, 2018. doi: 10.1142/S1793524518500997.

A. Owolanke, O. D. Ogwumu, T. Y. Kyagya, C. O. Okorie, dan G. I. Amakoromo, "A hybrid two-step method for direct solutions of general second order initial value problem," Engineering Mathematics Letters, vol. 2021, Article ID 1, Nov. 2020. doi: 10.28919/eml/4877.

A. O. Owolanke, O. Uwaheren, dan F. O. Obarhua, "An eight order two-step Taylor series algorithm for the numerical solutions of initial value problems of second order ordinary differential equations," Open Access Library Journal, vol. 4, p. e3486, 2017. doi: 10.4236/oalib.1103486.

O. K. Ogunbamike dan A. O. Owolanke, "Convergence of analytical solution of the initial-boundary value moving mass problem of beams resting on Winkler foundation," Electronic Journal of Mathematical Analysis and Applications, vol. 10, no. 1, pp. 129–136, 2022. doi: 10.21608/ejmaa.2022.311006.

O. K. Ogunbamike, I. T. Awolere, dan O. A. Owolanke, "Dynamic response of uniform cantilever beams on elastic foundation," African Journal of Mathematics and Statistics Studies, vol. 4, no. 1, pp. 47–62, 2021.

T. Oyedepo, A. K. Bello, J. U. Abubakar, A. M. Ayinde, dan T. F. Mohammed, "Numerical approximation of multi-order fractional differential equations by Galerkin method with Chebyshev polynomial basis," Journal of Fractional Calculus and Applications, vol. 15, no. 2, pp. 1–20, 2024. doi: 10.21608/jfca.2024.348330.

S. Sadiq, "Spectral Methods for Fractional Differential Equations," Disertasi Ph.D., School of Natural Sciences, National University of Sciences and Technology, 2024.

DOI: https://doi.org/10.37905/euler.v13i1.31034