jzs-10832

Numerical Treatment Solution of Volterra Integro-Fractional Differential Equation by Using Linear Spline Function

Karwan H.F. Jwamer 1, Shazad Sh. Ahmed 1 and  Diar Kh. Abdullah1*

1Department of Mathematics, College of Science, Sulaimani University, Sulaimani, Kurdistan Region, Iraq.

*Corresponding author:diar.khalid85@gmail.com

Original: 5 August 2020        Revised: 1 September 2020        Accepted: 26 September 2020        Published online: 20 December 2020  

Doi Link:

Abstract

In this article, we propose two new approximate methods based totally on the use of normal linear spline function and second employed with the Richardson Extrapolation technique the usage of discrete collocation points for approximating the solution of the Volterra integro-fractional differential equations (VIFDEs). The fractional derivatives are used in the Caputo sense. Illustrative examples are included to demonstrate the validity and applicability of the technique. A new technique with the resource of MatLab program is written to treat numerically VIFDEs using spline function, as well as, follow the Clenshaw Curtis rule for calculating the required integrals for those equations.

Key Words: Integro- fractional differential equation, Caputo derivative, Linear spline, Extrapolation method, Clenshaw  

References 

[1] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado. "Advances in Fractional Calculus". Netherlands:Springer. (2007).

[2] B. Guo, X. Pu, and F. Huang. "Fractional Partial Differential Equations and Their Numerical Solutions". World Scientific Publishing Co. Pte, Ltd. (2015).

[3] R. Almeida. "A Caputo fractional derivative of a function with respect to another function". Communications in Nonlinear Science and Numerical Simulation. Vol. 44, pp. 460-481. (2017). DOI.org/10.1016/j.cnsns.2016.09.006.

[4] I. Podlubny. "Fractional Differential Equations". Academic Press, USA. (1999).

[5] K. S. Miller. "An Introduction to Fractional Calculus and Fractional Differential Equations". J. Wiley and Sons, New York. (1993).

[6] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh. "A new definition of fractional derivative". Journal of Computational and Applied Mathematics. Vol. 264, pp. 65-70. (2014).

[7] G. -H.Gaoa, Z. -Zh. Sun, and H. -W. Zhang. "A new fractional numerical differentiation formula to approximate  The Caputo fractional derivative and its applications". Journal of Computational Physics. Vol. 259, pp. 33-50. (2014). DOI.org/10.1016 /j.jcp.2013.11.017.

[8] H. Mesgarani, H. Safdari, A. Ghasemian, and Y. Esmaeelzade. "The Cubic B-spline Operational Matrix Based on Haar Scaling Functions for Solving Varieties of the Fractional Integro-differential Equations". Journal of Mathematics. Vol. 51, No. 8, pp. 45-65. (2019).

[9] Sh. Sh. Ahmed, and Sh. A. Hama Salh. "Generalized taylor matrix method for solving linear integro-fractional Differential equations of volterra type". Applied Mathematical Sciences. Vol. 5, pp. 1765-1780. (2011).

[10] K. Maleknejad, M. N. Sahlan, and A. Ostadi. "Numerical solution of fractional integro-differential equation by using cubic B-spline wavelets". Proceedings of the World Congress on Engineering. Vol. 1. (2013).

[11] Sh. Sh. Ahmed, Sh.A.Hama Salh, and M.R.Ahmad. "Laplace adomian and laplace modified adomian decomposition methods for solving nonlinear integro-fractional differential equations of the volterra- hammerstein type". Iraqi Journal of Science. Vol. 60, pp. 2207-2222. (2019).

[12] Sh. Sh. Ahmed. "On system of linear volterra integro-fractional differential equations". Ph.D.dissertation,  University of Sulaimani, (2009).

[13] F. Tornabene, and N. Fantuzzi. "Mechanics of laminated composite doubly-curved shell structures". Bologna, Societa editrice Esculapio. (2014).

[14] R. B. Dash, and D. Das. "A Mixed Quadrature Rule by Blending Clenshaw-Curtis and Gauss-Legendre Quadrature Rules for Approximation of Real Definite Integrals in Adaptive". ( 2011).

[15] I. Faragó, Á. Havasi, and Z. Zlatev. "Efficient implementation of stable Richardson Extrapolation algorithms". Computers and Mathematics with Applications. Vol. 2010, pp. 2309-2325. (2010).  DOI.org/10.1016/j.camwa.2010.08.025.60.

[16] M. Zamani. "Three Simple Spline Method for Approximation and Interpolation of Data". Contemporary Engineering Sciences. Vol. 2, No. 8, pp. 373-381. (2009).

View Full Article