×

zbMATH — the first resource for mathematics

An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet. (English) Zbl 07241296
Summary: In this paper, Haar wavelet collocation technique is developed for the solution of Volterra and Volterra-Fredholm fractional integro-differential equations. The Haar technique reduces the given equations to a system of linear algebraic equations. The derived system is then solved by Gauss elimination method. Some numerical examples are taken from literature for checking the validation and convergence of the proposed method. The maximum absolute errors are compared with the exact solution. The maximum absolute and mean square root errors for different number of collocation points are calculated. The results show that Haar method is efficient for solving these equations. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is described in the Caputo sense. All algorithms for the developed method are implemented in MATLAB (R2009b) software.

MSC:
65R20 Numerical methods for integral equations
47G20 Integro-differential operators
Software:
Matlab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Katugampola, U. N., New approach to a generalized fractional integral, Appl. Math. Comput., 218, 860-865 (2011) · Zbl 1231.26008
[2] Rawashded, E. A., Legendre wavelets method for fractional integro-differential equations, Appl. Math. Sci., 5, 50, 2467-2474 (2011) · Zbl 1250.65160
[3] Saleh, M. H.; Amer, S. M.; Mohamed, M. A.; Abdelrhman, N. S., Approximate solution of fractional integro-differential equations by Taylor expansion and Legendre wavelets methods, Cubo, 15, 3, 89-103 (2013) · Zbl 1292.65143
[4] Arikoglu, A.; Okozol, I., Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40, 2, 521-529 (2009) · Zbl 1197.45001
[5] Mittal, R. C.; Nigam, R., Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech., 4, 2, 87-94 (2008)
[6] Vanani, S. K.; Amnataei, A., Operational Tau approximation for a general class of fractional integro-differential equations, Comput. Math. Appl., 30, 3, 655-674 (2011) · Zbl 1247.65174
[7] Yang, C.; Hou, J., Numerical solution of integro-differential equations of fractional order by Laplace decomposition method, WSEAS Trans. Math., 12, 1173-1183 (2013)
[8] Mohammed, D. S., Numerical solution of fractional integro-differential equations by least squares and shifted Chebyshev polynomial methods, Math. Probl. Eng., 1, 8, 1-5 (2014) · Zbl 1407.65075
[9] S. Mahdy, A. M.; Shwayyea, R. T., Numerical solution of fractional integro-differential equations by least squares and shifted Laguerre polynomials Pseudo spectral methods, Int. J. Sci. Eng. Res., 7, 7, 1589-1596 (2016)
[10] Hamoud, A.; Ghadle, K. P., Modified Laplace decomposition method for fractional Volterra Fredholm integro-differential equations, J. Math. Model., 6, 1, 91-104 (2018) · Zbl 1413.65482
[11] Jani, M.; Bhatta, D.; Javadi, S., Numerical solution of fractional integro-differential equations with nonlocal conditions, Appl. Appl. Math., 12, 1, 98-111 (2017) · Zbl 1368.65125
[12] Hamoud, A.; Gadle, K.; Atsha, S., The approximate solutions of fractional integro-differential equations by usig modiffied Adomain decomposition method, Khayyam J. Math., 5, 1, 39-57 (2018)
[13] Yi, M.; Huang, J.; Wang, L., Operational matrix method for solving variable order fractional integro-differential equations, Khayyam J. Math., 96, 5, 361-377 (2013) · Zbl 1356.65205
[14] Alkan, S.; Hatipoglu, V. F., Approximate solutions of Volterra Fredholm integro-differential equations of fractional order, Tbilisi Math. J., 10, 2, 1-13 (2017) · Zbl 1360.65301
[15] Elbeleze, A.; Kilicman, A.; Taib, B. M., Approximate solution of integro-differential equations of fractional of arbitrary order, J. King Saud Univ. Sci., 28, 61-68 (2016)
[16] Ma, X.; Huang, C., Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math. Comput., 219, 6750-6760 (2013) · Zbl 1290.65130
[17] Setia, A.; Liu, Y.; Vatsala, A. S., Numerical solution of Fredholm-bernstein polynomials Volterra fractional integro-differential equations wth nonlocal boundary conditions, J. Fract. Calc. Appl., 5, 2, 155-165 (2014)
[18] Khader, M. M.; Sweilam, N. H., On the approximate solutions for system of fractional integro-differential equations using Chebyshev Pseudo spectral method, J. Fract. Calc. Appl., 37, 24, 9819-9828 (2013) · Zbl 1427.65419
[19] Keshavarz, E.; Ordokhani, Y.; Razzaghi, M., Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Model., 38, 24, 6038-6051 (2014) · Zbl 1429.65170
[20] Wang, Y.; Zhu, L.; Wang, Z., Fractional order Euler functions for solving fractional integro-differential equations with weakly singular kernel, Adv. Differential Equations, 254, 2, 436-456 (2018)
[21] Babaei, A.; Jafari, H.; Banihashemi, S., Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, J. Comput. Appl. Math., 112908 (2020) · Zbl 07195350
[22] Benyoussef, S.; Rahmoune, A., Efficient spectral-collocation methods for a class of linear Fredholm integro-differential equations on the half-line, J. Comput. Appl. Math., 112894 (2020) · Zbl 07195347
[23] Lepik, Ü., Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comput., 176, 1, 324-333 (2006) · Zbl 1093.65123
[24] Aziz, I.; Amin, R., Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet, Appl. Math. Model., 40, 10286-10299 (2016) · Zbl 1443.65089
[25] Khashan, M. M.; Amin, R.; Syam, M. I., A new algorithm for fractional Riccati type differential equations by using Haar wavelet, Mathematics, 7, 6, 545 (2019)
[26] Amin, R.; Nazir, S.; Magariño, I. G., Efficient sustainable algorithm for numerical solution of nonlinear delay Fredholm-Volterra integral equations via Haar wavelet for dense sensor networks in emerging telecommunications, Trans. Emerg. Telecommun. Technol., Article e3877 pp. (2020)
[27] Amin, R.; Nazir, S.; Magariño, I. G., A collocation method for numerical solution of nonlinear delay integro-differential equations for wireless sensor network and internet of things, Sensors, 20 (2020)
[28] Dabiri, A.; Moghaddam, B. P.; Machado, J. T., Optimal variable-order fractional PID controllers for dynamical systems, J. Comput. Appl. Math., 339, 40-48 (2018) · Zbl 1392.49033
[29] Babaei, A.; Moghaddam, B.; Banihashemi, S.; Machado, J. A., Numerical solution of variable-order fractional integro-partial differential equations via sinc collocation method based on single and double exponential transformations, Commun. Nonlinear Sci. Numer. Simul., 82, 104985 (2020)
[30] Moghaddam, B. P.; Dabiri, A.; Machado, J. A.T., Application of variable-order fractional calculus in solid mechanics, (Applications in Engineering, Life and Social Sciences, Part A (2019), De Gruyter), 207-224
[31] Rawashdeh, E. A., Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput., 176, 1, 1-6 (2006) · Zbl 1100.65126
[32] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002
[33] Lepik, U., Solving fractional integral equations by the Haar wavelet method, Appl. Math. Comput., 214, 468-478 (2009) · Zbl 1170.65106
[34] Majak, J.; Shvartsman, B.; Karjust, K.; Mikola, M.; Haavajoe, A.; Pohlak, M., On the accuracy of the haar wavelet discretization method, Composites B, 80, 321-327 (2015)
[35] Alkan, S., A numerical method for solution of integro-differential equations of fractional order, Sar. Univ. J. Sci., 21, 2, 82-89 (2017)
[36] Zhou, F.; Xu, X., Numerical solution of fractional Volterra Fredholm integro-differential equations with mixed boundary conditions via Chebyshev wavelet method, Int. J. Comput. Math., 96, 2, 436-456 (2018)
[37] Yi, M.; Wang, L.; Huang, J., Legendre wavelets method for the numerical solution of fractional integro-differential equations with weakly singular kernel, Int. J. Comput. Math., 40, 3422-3437 (2016) · Zbl 07159260
[38] Zill, G. D., Differential Equations with Boundary-Value Problems (2016), Nelson Education
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.