A numerical method based on Chelyshkov polynomials for solving fractional integro-differential equations

Subject Areas : Statistics

Reza Dehghan ¹

1 - Department of Mathematics, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran

Received: 2018-08-04 Accepted : 2021-01-18 Published : 2021-03-01

Keywords: مشتق کسری کپوتو, حسابان کسری, معادلات دیفرانسیل-انتگرال ولترا, چندجمله ایهای متعامد,

Abstract :

In this paper, Chelyshkov expansion approach is presented for solving Volterra fractional order integro-differential equations with Caputo derivative. By means of the properties of Chelyshkov polynomials and numerical integral formula , the solution of fractional integro-differential equations reduced to the solution of algebraic equations. Then, by solving the system of algebraic equations, the solution of the differential-integral equation is presented as a function in the terms of Chelyshkov polynomials. Accuracy and error analysis have been investigated and since the accuracy of the obtained results for fractional integro-differential equations depends on the number of selected Chelyshkov polynomials therefore, with the increase in the number of Chelyshkov polynomials, we can achieve desirable accuracy step by step. All calculations are done by MATLAB software. Also, the numerical results of based on Chelyshkov polynomials method are compared with the results of some of the available methods for the validity, accuracy and efficiency of the technique.

References:

[1] Saadatmandi, A., and Dehghan, M., A Legendre collocation method for fractional integro-differential equations, Journal of Vibration and Control, 17(13) (2011) 2050–2058.

 

[2] Baillie, R.T., Long memory processesand fractional integration in econometrics. Journal of Econometrics, 73 (1996) 5–59.

 

[3] Bohannan, G.W., Analog fractional order controller in temperature and motor control applications. Journal of Vibration and Control., 14 (2008)1487–1498.

 

[4] Debnath, L., Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences., (2003) 3413–3442.

 

[5] Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri A and Mainardi F (eds) Fractals and Fractional Calculus in Continuum Mechanics. New York: Springer-Verlag., (1997) 291–348.

 

[6] Mohammadi, F., Moradi, L., Numerical treatment of fractional-order nonlinear system of delay integro- differential equations arising in biology., Asian-European Journal of Mathematics., 12(1) (2019) DOI: 10.1142/ S1793557119500682

 

[7] Momani, S., Noor, M.A., Numerical  methods for fourthorder fractional integro-differential equations, Appl, Math. Comput., 182 (2006) 754-760.

 

[8] Hashim, I., Adomian decomposition method for solving BVPs for fourth-order integro-differential equations, J.Comput. Appl. Math., 193 (2006) 658-664.

[9] Neamaty, A., Agheli, B., Darzi, R., Numerical solution of high-order fractional Volterra integro-differential equations by variational homotopy perturbution iteration method, Journal of Computational and Nonlinear Dynamics.,(2015) doi:10.1115/1.4030062

 

[10] S. Karimi Vanani, A. Aminataei., Operational Tau approximation for a general class of fractional integro-differential equations, Computational & Applied Mathematics., 30 (3) (2011) 655-674.

 

[11] Inc, M., The approximate and exact  solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, Journal of Mathematical Analysis and Applications 345 (2008) 476–484.

 

[12] Al-Jamal, M.F., Rawashde, E.A., The Approximate Solution of Fractional Integro- Differential Equations. Int. J. Contemp. Math. Sciences, 4 (2009) 1067–1078.

 

[13] Nazari,  D., Shahmorad, S., Application of the fractional differential transform method to fractional–order integro–differential equations with nonlocal boundary conditions., Journal of Computational and Applied Mathematics., 234 (2010)  883–891.

 

[14] Pedas, A., Tamme, E., Spline collocation method for integro-differential equations with weakly singular kernels. Journal of Computational and Applied Mathematics, 197 (2006)  253–269.

 

[15] Podlubny, I., Fractional differential equations. Academic Press, San Diego., (1999)

 

[16] Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y., Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Applied Mechanics and Engineering., 194 (2005) 743–773.

 

[17] Chelyshkov, V.S., Alternative orthogonal polynomials and quadratures., Electron. Trans. Numer. Anal. 25 (7) (2006) 17–26.