This article presents a novel numerical method for seeking the numerical solutions of fractional order differential equations using hybrid functions consisting of block-pulse functions and Taylor polynomials. The fractional integrals operational matrix of the hybrid function is conducted through projecting the hybrid functions onto block-pulse functions. Then, the fractional order differential equations are converted to a set of algebraic equations via the derived operational matrix. Then, the numerical solutions are obtained via solving the algebraic equations. Moreover, we perform error analysis of the algorithm and gives the upper bound of absolute error. Finally, numerical examples are given to show the effectiveness of the proposed method.
Citation: Hailun Wang, Fei Wu, Dongge Lei. A novel numerical approach for solving fractional order differential equations using hybrid functions[J]. AIMS Mathematics, 2021, 6(6): 5596-5611. doi: 10.3934/math.2021331
This article presents a novel numerical method for seeking the numerical solutions of fractional order differential equations using hybrid functions consisting of block-pulse functions and Taylor polynomials. The fractional integrals operational matrix of the hybrid function is conducted through projecting the hybrid functions onto block-pulse functions. Then, the fractional order differential equations are converted to a set of algebraic equations via the derived operational matrix. Then, the numerical solutions are obtained via solving the algebraic equations. Moreover, we perform error analysis of the algorithm and gives the upper bound of absolute error. Finally, numerical examples are given to show the effectiveness of the proposed method.
| [1] |
V. Lakshmikantham, A. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. Theor., 69 (2008), 2677–2682. doi: 10.1016/j.na.2007.08.042
|
| [2] |
F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelaticity: An experimental study, Commun. Nonlinear Sci., 15 (2010), 939–945. doi: 10.1016/j.cnsns.2009.05.004
|
| [3] |
J. Wang, Realizations of generalized warburg impedance with RC ladder networks and transmission lines, J. Electrochem. Soc., 134 (1987), 1915. doi: 10.1149/1.2100789
|
| [4] |
L. L. Huang, J. H. Park, G. C. Wu, Z. W. Mo, Variable-order fractional discrete-time recurrent neural networks, J. Comput. Appl. Math., 370 (2020), 112633. doi: 10.1016/j.cam.2019.112633
|
| [5] |
G. C. Wu, M. Luo, L. L. Huang, S. Banerjee, Short memory fractional differential equations for new memristor and neural network design, Nonlinear Dynam., 100 (2020), 3611–3623. doi: 10.1007/s11071-020-05572-z
|
| [6] |
G. C. Wu, M. Niyazi Çankaya, S. Banerjee, Fractional q-deformed chaotic maps: A weight function approach, Chaos, 30 (2020), 121106. doi: 10.1063/5.0030973
|
| [7] | T. U. Khan, M. A. Khan, Y. M. Chu, A new generalized Hilfer-type fractional derivative with applications to space-time diffusion equation, Results Phys., 22 (2021), 103953. |
| [8] |
M. F. El Amin, Derivation of fractional-derivative models of multiphase fluid flows in porous media, J. King Saud. Univ. Sci., 33 (2021), 101346. doi: 10.1016/j.jksus.2021.101346
|
| [9] | H. R. Marzban, A new fractional orthogonal basis and its application in nonlinear delay fractional optimal control problems, ISA T., 2020, In press. |
| [10] |
Y. Liu, Y. Du, H. Li, S. He, W. Gao, Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction diffusion problem, Comput. Math. Appl., 70 (2015), 573–591. doi: 10.1016/j.camwa.2015.05.015
|
| [11] |
V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictor-corrector method for fractional differential equations, Appl. Math. Comput., 244 (2014), 158–182. doi: 10.1016/j.amc.2014.06.097
|
| [12] |
C. Li, Y. Wang, Numerical algorithm based on adomian decomposition for fractional differential equations, Comput. Math. Appl., 57 (2009), 1672–1681. doi: 10.1016/j.camwa.2009.03.079
|
| [13] |
G. C. Wu, E. Lee, Fractional variational iteration method and its application, Phys. Lett. A, 374 (2010), 2506–2509. doi: 10.1016/j.physleta.2010.04.034
|
| [14] |
B. Ghazanfari, F. Veisi, Homotopy analysis method for the fractional nonlinear equations, J. King Saud. Univ. Sci., 23 (2011), 389–393. doi: 10.1016/j.jksus.2010.07.019
|
| [15] | H. Dehestani, Y. Ordokhani, M. Razzaghi, Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations, Engineering with Computers, (2020), 1–16. |
| [16] |
S. Najafalizadeh, 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. doi: 10.1016/j.cam.2017.05.039
|
| [17] |
M. H. Alshbool, O. Isik, I. Hashim, Fractional bernstein series solution of fractional diffusion equations with error estimate, Axioms, 10 (2021), 6. doi: 10.3390/axioms10010006
|
| [18] |
J. R. Loh, C. Phang, Numerical solution of fredholm fractional integro-differential equation with right-sided caputo's derivative using bernoulli polynomials operational matrix of fractional derivative, Mediterr J. Math., 16 (2019), 1–25. doi: 10.1007/s00009-018-1275-9
|
| [19] |
İ. Avcı, N. I. Mahmudov, Numerical solutions for multi-term fractional order differential equations with fractional Taylor operational matrix of fractional integration, Mathematics, 8 (2020), 96. doi: 10.3390/math8010096
|
| [20] |
S. Mashayekhi, M. Razzaghi, Numerical solution of nonlinear fractional integro-differential equations by hybrid functions, Eng. Anal. Bound. Elem., 56 (2015), 81–89. doi: 10.1016/j.enganabound.2015.02.002
|
| [21] |
B. Yuttanan, M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model., 70 (2019), 350–364. doi: 10.1016/j.apm.2019.01.013
|
| [22] |
M. H. Heydari, Chebyshev cardinal wavelets for nonlinear variable-order fractional quadratic integral equations, Appl. Numer. Math., 144 (2019), 190–203. doi: 10.1016/j.apnum.2019.04.019
|
| [23] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1998. |
| [24] |
H. Marzban, M. Razzaghi, Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series, J. Sound. Vib., 292 (2006), 954–963. doi: 10.1016/j.jsv.2005.08.007
|
| [25] |
A. Kilicman, Z. A. A. A. Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput., 187 (2007), 250–265 doi: 10.1016/j.amc.2006.08.122
|
| [26] | E. Kreyszig, Introductory functional analysis with applications, New York: Wiley, 1978. |
| [27] |
S. K. Damarla, M. Kundu, Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices, Appl. Math. Comput., 263 (2015), 189–203. doi: 10.1016/j.amc.2015.04.051
|
| [28] |
Z. Odibat, S. Momani, Modified Homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos Soliton. Fract., 36 (2008), 167–174. doi: 10.1016/j.chaos.2006.06.041
|
| [29] |
Y. Li, Solving a nonlinear fractional differential equation using chebyshev wavelets, Commun. Nonlinear Sci., 15 (2010), 2284–2292. doi: 10.1016/j.cnsns.2009.09.020
|
| [30] |
A. El-Mesiry, A. El-Sayed, H. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput., 160 (2005), 683–699. doi: 10.1016/j.amc.2003.11.026
|
Figures(2) / Tables(3)
Hailun Wang, Fei Wu, Dongge Lei. A novel numerical approach for solving fractional order differential equations using hybrid functions[J]. AIMS Mathematics, 2021, 6(6): 5596-5611. doi: 10.3934/math.2021331
DownLoad: