## Articulo

• Similares en SciELO

## versión impresa ISSN 0716-0917

### Proyecciones (Antofagasta) vol.39 no.1 Antofagasta feb. 2020

#### http://dx.doi.org/10.22199/issn.0717-6279-2020-01-0013

Artículos

A new approach for solving linear fractional integro-differential equations and multi variable order fractional differential equations

1 Kashmar Higher Education Institute, Dept. of Mathematics, Kashmar, Iran. E-mail: f.ghomanjani@kashmar.ac.ir

Abstract

In the sequel, the numerical solution of linear fractional integrodifferential equations (LFIDEs) and multi variable order fractional differential equations (MVOFDEs) are found by Bezier curve method (BCM) and operational matrix. Some numerical examples are stated and utilized to evaluate the good and accurate results.

Keywords: Fractional integro-differential equations; Bezier curve; Variable order fractional differential equation; Caputo’s variable order fractional derivative

Acknowledgement.

The author would like to thank the anonymous reviewers of this paper for their careful reading, constructive comments and nice suggestions which have improved the paper very much.

[1] A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals”, Boundary value problems, Art. ID. 62, Dec. 2012, doi: 10.1186/1687-2770-2012-62. [ Links ]

[2] F. Ghomanjani, M. H. Farahi, and M. Gachpazan, “Bezier control points method to solve constrained quadratic optimal control of time varying linear systems”, Computational and applied mathematics, vol. 31, no. 3, pp. 1-24, 2012, doi: 10.1590/S1807-03022012000300001. [ Links ]

[3] F. Ghomanjani andM. H. Farahi , “The Bezier control points method for solving delay differential equation”, Intelligent control and automation, vol. 3, no. 2, pp. 188-196, 2012, doi: 10.4236/ica.2012.32021. [ Links ]

[4] F. Ghomanjani and M. Hadifarahi, “Optimal Control of Switched Systems based on Bezier Control Points”, International journal of intelligent systems and applications, vol. 4, no. 7, pp. 16-22, Jun. 2012, doi: 10.5815/ijisa.2012.07.02. [ Links ]

[5] F. Ghomanjani , M. H. Farahi , and A. V. Kamyad, “Numerical solution of some linear optimal control systems with pantograph delays”, IMA journal of mathematical control and information, vol. 32, no. 2, pp. 225-243, Jun. 2015, doi: 10.1093/imamci/dnt037. [ Links ]

[6] F. Ghomanjani , E. Khorram, “Approximate solution for quadratic Riccati differential equation”, Journal of Taibah university for science, vol. 11, no. 2, pp. 246-250, Mar. 2017, doi: 10.1016/j.jtusci.2015.04.001. [ Links ]

[7] F. Ghomanjani , “A new approach for solving fractional differential algebraic equations”, Journal of Taibah university for science , vol. 11, no. 6, pp. 1158-1164, Nov. 2017, doi: 10.1016/j.jtusci.2017.03.006. [ Links ]

[8] H. Khalil, R. A. Khan, and M. M. Rashidi, “Brenstien polynomials and applications to fractional differential equations”, Computational methods for differential equations, vol. 3, no. 1, pp. 14-35, 2015. [On line]. Available: https://bit.ly/2OjgcMYLinks ]

[9] D. S. Mohammed, “Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomial”, Mathematical problems in engineering, vol. 2014, pp. 1-5, 2014, doi: 10.1155/2014/431965. [ Links ]

[10] P. Rahimkhani, Y. Ordokhani, E. Babolian, “Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet”, Journal of Computational and applied mathematics , vol. 309, Jan. 2017, doi: 10.1016/j.cam.2016.06.005. [ Links ]

[11] J. Zheng, T. W. Sederberg, and R. W. Johnson, “Least squares methods for solving differential equations using Bezier control points”, Applied numerical mathematics, vol. 48, no. 2, pp. 237-252, Feb. 2004, doi: 10.1016/j.apnum.2002.01.001. [ Links ]

[12] L. E. S. Ramírez and C. F M. Coimbra, “On the variable order dynamics of the nonlinear wake caused by a sedimenting particle”. Physica D: nonlinear phenomena, vol. 240, no. 13, pp. 1111-1118, Jun. 2011, doi: 10.1016/j.physd.2011.04.001. [ Links ]

[13] H. G. Sun, W. Chen, H. Wei, and Y. Q. Chen, “A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems”, The european physical journal special topics, vol. 193, Art. ID. 185, Mar. 2011, doi: 10.1140/epjst/e2011-01390-6. [ Links ]

[14] L. Zhu and Q. Fan, “Solving fractional nonlinear Fredholm integrodifferential equations by the second kind Chebyshev wavelet”, Communications in nonlinear science and numerical simulation, vol. 17, no. 6, pp. 2333-2341, Jun. 2012, doi: 10.1016/j.cnsns.2011.10.014. [ Links ]

[15] M. S. M. Selvi and G. Hariharan, “Wavelet-Based analytical algorithm for solving steady-state concentration in immobilized glucose isomerase of Packed-Bed Reactor Model”, The Journal of membrane biology, vol. 249, no. 4, pp. 1-10, Aug. 2016, doi: 10.1007/s00232-016-9905-2. [ Links ]

[16] A. K. Gupta and S. S. Ray, “Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method”, Applied mathematical modelling, vol. 39, no. 17, pp. 5121-5130, Sep. 2015, doi: 10.1016/j.apm.2015.04.003. [ Links ]

[17] M. H. Heydari, M. R. Hooshmandasl, C. Cattani, andG. Hariharan , “An optimization Wavelet method for multi variable-order fractional differential equations”, Fundamenta informaticae, vol. 151, no. 1-4, pp. 255-273, Mar. 2017, doi: 10.3233/FI-2017-1491. [ Links ]

Received: February 2019; Accepted: May 2019