Hat functions operational matrix for solving the nonlinear fractional-order integro-differential equation

Khajehnasiri, Amir Ahmad

doi:10.30495/maca.2024.720828

Hat functions operational matrix for solving the nonlinear fractional-order integro-differential equation

10.30495/maca.2024.720828

Volume 6, Issue 4
Autumn 2024

Pages 59-69

Mathematical Analysis and its Contemporary Applications

Document Type : Original Article

Author

Amir Ahmad Khajehnasiri ¹^{, 2}

¹ Department of Economics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

² Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

This paper aims to construct a general formulation for the hat basis functions of the fractional integral operator. We derive the hat basis functions operational matrix of the fractional order integration, and use it to solve the fractional-order integro-differential equation. The method is described and illustrated with a numerical example. The results show that the method is accurate and easy to apply.

Keywords

Hat basis functions

Operational matrix

Block pulse function

Fractional calculus

Volterra Fredholm

[1] L. Gaul, P. Klein, and S. Kempfle, Damping description involving fractional operators, Mech. Syst. Signal Process, 5 (1991), 81–88.

[2] W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68 (1995), 46–53.

[3] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and C. Cattani. A computational method for solving stochastic Ito–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys., 270 (2014), 402–415.

[4] E. Babolian and M. Mordad, A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions, Comput. Math. Appl., 62 (2011), 187–198.

[5] S. Najafalizadeh and R. Ezzati, A block pulse operational matrix method for solving two-dimensional nonlinear integro-differential equations of fractional order, J. Comput. Appl. Math., 326 (2017) 159-170.

[6] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and C. Cattani, An efficient computational method for solving nonlinear stochastic Ito integral equations: Application for stochastic problems in physics, J. Comput. Phys., 283 (2015), 148–168.

[7] F. Mirzaee and E. Hadadiyan, Application of two-dimensional hat functions for solving space-time integral equations, J. Appl. Math. Comput., 4 (2015), 1–34.

[8] M.P. Tripathi, V. K. Baranwal, R. K. Pandey, and O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1327–1340.

[9] A. Ebadian, H. Rahmani Fazli, and A. A. Khajehnasiri. Solution of nonlinear fractional diffusion-wave equation by triangular functions, SeMA J., 72 (2015), 37–46.

[10] H. Rahmani Fazli, F. Hassani, A. Ebadian, and A. A. Khajehnasiri, National economies in state-space of fractional-order financial system, Afr. Mate., 10 (2015), 1–12.

[11] A. Ebadian and A. A. Khajehnasiri, Block-pulse functions and their applications to solving systems of higher-order nonlinear Volterra integro-differential equations, Electron. J. Diff. Equ., 54 (2014), 1–9.

[12] H. Saeedi, N. Mollahasani, M. M. Moghadam, and G. N. Chuev, An operational haar wavelet method for solving fractional Volterra integral equations, Int. J. Appl. Math. Comput. Sci., 21 (2011), 535–547.

[13] M. Saeedi and M.M. Moghadam. Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS Wavelets, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 1216–1226.

[14] E.A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176 (2006), 1–6.

[15] S. Mashayekbi and M. Razzaghi, Numerical solution of nonlinear fractional integro-differential equation by hybrid functions, Engin. Anal. Bound. Elements, 56 (2015), 81–89.

[16] A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40 (2009), 521–529.

[17] A. Kilicman and Z. A. Al Zhour. Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput., 187 (2007) 250–265.

[18] S. Momani and M. A. Noor, Numerical methods for fourth-order fractional integro-differential equations, Appl. Math. Comput., 182 (2006), 754–760.

[19] P. N. Paraskevopoulos, Legendre series approach to identification and analysis of linear systems, IEEE Trans. Automat. Control, 30(6) (1985), 585–589.

[20] C. Hwang and Y. P. Shih, Parameter identification via Laguerre polynomials, Internat, J. Systems Sci, 13 (1982) 209-17.

[21] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336.

[22] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62 (2011), 2364–2373.

XML

PDF 393.01 K