Document Type : Research Paper

**Authors**

Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

**Abstract**

In this paper, we propose a new numerical algorithm for the approximate solution of non-homogeneous fractional differential equation. Using this algorithm the fractional differential equations are transformed into a system of algebraic linear equations by operational matrices of block-pulse and hybrid functions. Based on our new algorithm, this system of algebraic linear equations can be solved by a proposed (TSI) method. Further, some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm.

**Keywords**

[1] H. Aminikhah, A. H. Refahi Sheikhani, and H. Rezazadeh, Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method, Bol. Soc. Paran. Mat., 34 (2016), 213-229.

[2] H. Aminikhah, A. R . Sheikhani, and H. Rezazadeh, Exact solutions for the fractional differential equations by using the first integral method, Nonlinear Engineering, 4 (2015), 15–22.

[3] H. Aminikhah, A. H. Refahi Sheikhani, and H. Rezazadeh, Stability analysis of linear distributed order system with multiple time delays, U.P.B. Sci. Bull, 77 (2015), 207–218.

[4] A. Ansari, and A. H. Refahi Sheikhani. New identities for the wright and the mittag-leffler functions using the laplace transform, Asian-European Journal of Mathematics, 7 (2014), 1–8.

[5] A. Ansari, A. H. Refahi Sheikhani, and H. Saberi Najafi, Solution to system of partial fractional differential equations using the fractional exponential operators, Math. Meth. Appl. Sci., 35 (2012), 119-123.

[6] A. Ansari, A. Refahi Sheikhani, and S. Kordrostami, On the generating function e xt+yφ(t) and its fractional calculus, Cent. Eur. J. Phys., 11 (2013), 1457–1462.

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[9] P. J. Lanzkron, D. J. Rose, and D. B. Szyld, Convergence of nested iterative methods for linear systems, Numer. Math., 58 (1991), 685–702.

[10] A. C. J. Luo, Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, Springer, (2010).

[2] H. Aminikhah, A. R . Sheikhani, and H. Rezazadeh, Exact solutions for the fractional differential equations by using the first integral method, Nonlinear Engineering, 4 (2015), 15–22.

[3] H. Aminikhah, A. H. Refahi Sheikhani, and H. Rezazadeh, Stability analysis of linear distributed order system with multiple time delays, U.P.B. Sci. Bull, 77 (2015), 207–218.

[4] A. Ansari, and A. H. Refahi Sheikhani. New identities for the wright and the mittag-leffler functions using the laplace transform, Asian-European Journal of Mathematics, 7 (2014), 1–8.

[5] A. Ansari, A. H. Refahi Sheikhani, and H. Saberi Najafi, Solution to system of partial fractional differential equations using the fractional exponential operators, Math. Meth. Appl. Sci., 35 (2012), 119-123.

[6] A. Ansari, A. Refahi Sheikhani, and S. Kordrostami, On the generating function e xt+yφ(t) and its fractional calculus, Cent. Eur. J. Phys., 11 (2013), 1457–1462.

[7] Z. Z. Bai, The convergence of the two-stage iterative method for Hermitian positive definite linear systems, Appl. Math. Lett., 11 (1998), 1–5.

[8] A. Deb, and S. Ghosh, Power electronic Systems: Walsh analysis With Matlab. CRC Press, Taylor and Francis Group, LLC., 2014.

[9] P. J. Lanzkron, D. J. Rose, and D. B. Szyld, Convergence of nested iterative methods for linear systems, Numer. Math., 58 (1991), 685–702.

[10] A. C. J. Luo, Dynamical Systems: Discontinuity, Stochasticity and Time-Delay, Springer, (2010).

[11] M. Meerschaert, and C. Tadjeran, Finite difference approximations for two-sided spacefractional partial differential equations, Applied Numerical Mathematics, 56 (2006), 80–90.

[12] N. K. Nichols, On the convergence of two-stage iterative processes for solving linear equations, SIAM Journal on Numerical Analysis, 10 (1973), 460–469.

[13] A. Patra, and G. P. Rao, General Hybrid Orthogonal Functions and Their Applications in Systems and Control, Springer, LNCIS 213, London, 1996. 207–218.

[14] I. Podlubny, Fractional differential equations. Mathematics in Science and Engineering, Academic Press, New York, NY, USA., 1999.

[15] A. H. Refahi Sheikhani, A. Ansari, H. Saberi Najafi, and F. Merhdoust, Analytic study on linear systems of distributed order fractional differential equations, Le Matematiche, 67 (2012), 3-13.

[16] E. Reyes Melo, J. Martinez Vega, C. Guerrero Salazar, and U. Ortiz Mendez, Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials, Journal of Applied Polymer Science, 98 (2005), 923-935.

[17] H. Rezazadeh, H. Aminikhah, and A. H. Refahi Sheikhani, Stability analysis of Hilfer fractional differential systems, Math. Commun., 21 (2016), 45-64.

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

[19] H. Saberi Najafi, A. Refahi Sheikhani, and A. Ansari, Stability analysis of distributed order fractional differential equations, Abstract and Applied Analysis, 2011(2011), 1-12.

[20] H. Saberi Najafi, and A. H. Refahi Sheikhani, FOM-inverse vector iteration method for computing few smallest (largest) eigenvalues of pair (A, B), Applied mathematics and computations, 188 (2007), 641–647.

[21] H. Saberi Najafi, S. A. Edalatpanah, and A. H. Refahi Sheikhani, Convergence Analysis of Modified Iterative Methods to Solve Linear Systems, Mediterranean Journal of Mathematics, 11(2014), 1019–1032.

[22] H. Saberi Najafi, A. H. Refahi Sheikhani, and M. Akbari, Weighted FOM-inverse vector iteration method for computing a few smallest (largest) eigenvalues of pair (A, B), Applied mathematics and computation, 192 (2007), 239–246.

[23] R. C. Soni and D. Singh, Some Theorems Connecting the Unified Fractional Integral Operators and the Laplace Transform, K. M. J., 45 (2005), 153–159.

[24] D. Valerio, J. J. Trujillo, M. Rivero, J. T. Machado, and D. Baleanu, Fractional calculus: a survey of useful formulas, The European Physical Journal Special Topics, 222 (2013), 1827-1846.

[25] C. Yang, Numerical Solution of Nonlinear Fredholm Integro differential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials, Mathematical Problems in Engineering, 2011 (2011), 1–12.

[26] M. Yi, J. Huang, and J. Wei, block-pulse operational matrix method for solving fractional partial differential equation, Applied Mathematics and Computation., 221 (2013), 121–131.

[12] N. K. Nichols, On the convergence of two-stage iterative processes for solving linear equations, SIAM Journal on Numerical Analysis, 10 (1973), 460–469.

[13] A. Patra, and G. P. Rao, General Hybrid Orthogonal Functions and Their Applications in Systems and Control, Springer, LNCIS 213, London, 1996. 207–218.

[14] I. Podlubny, Fractional differential equations. Mathematics in Science and Engineering, Academic Press, New York, NY, USA., 1999.

[15] A. H. Refahi Sheikhani, A. Ansari, H. Saberi Najafi, and F. Merhdoust, Analytic study on linear systems of distributed order fractional differential equations, Le Matematiche, 67 (2012), 3-13.

[16] E. Reyes Melo, J. Martinez Vega, C. Guerrero Salazar, and U. Ortiz Mendez, Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials, Journal of Applied Polymer Science, 98 (2005), 923-935.

[17] H. Rezazadeh, H. Aminikhah, and A. H. Refahi Sheikhani, Stability analysis of Hilfer fractional differential systems, Math. Commun., 21 (2016), 45-64.

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

[19] H. Saberi Najafi, A. Refahi Sheikhani, and A. Ansari, Stability analysis of distributed order fractional differential equations, Abstract and Applied Analysis, 2011(2011), 1-12.

[20] H. Saberi Najafi, and A. H. Refahi Sheikhani, FOM-inverse vector iteration method for computing few smallest (largest) eigenvalues of pair (A, B), Applied mathematics and computations, 188 (2007), 641–647.

[21] H. Saberi Najafi, S. A. Edalatpanah, and A. H. Refahi Sheikhani, Convergence Analysis of Modified Iterative Methods to Solve Linear Systems, Mediterranean Journal of Mathematics, 11(2014), 1019–1032.

[22] H. Saberi Najafi, A. H. Refahi Sheikhani, and M. Akbari, Weighted FOM-inverse vector iteration method for computing a few smallest (largest) eigenvalues of pair (A, B), Applied mathematics and computation, 192 (2007), 239–246.

[23] R. C. Soni and D. Singh, Some Theorems Connecting the Unified Fractional Integral Operators and the Laplace Transform, K. M. J., 45 (2005), 153–159.

[24] D. Valerio, J. J. Trujillo, M. Rivero, J. T. Machado, and D. Baleanu, Fractional calculus: a survey of useful formulas, The European Physical Journal Special Topics, 222 (2013), 1827-1846.

[25] C. Yang, Numerical Solution of Nonlinear Fredholm Integro differential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials, Mathematical Problems in Engineering, 2011 (2011), 1–12.

[26] M. Yi, J. Huang, and J. Wei, block-pulse operational matrix method for solving fractional partial differential equation, Applied Mathematics and Computation., 221 (2013), 121–131.

July 2021

Pages 659-669

**Receive Date:**07 October 2018**Revise Date:**17 January 2019**Accept Date:**25 February 2019**First Publish Date:**01 July 2021