Wavelet‎-based numerical ‎method‎ ‎‎‎‎for solving fractional integro-differential equation with a weakly singular ‎kernel

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎University of ‎Hormozgan‎, ‎P‎. ‎O‎. ‎Box 3995‎, ‎Bandarabbas‎, ‎Iran

2 Department of Biomedical Sciences and Morphological and Functional Imaging‎,‎ University of Messina‎, ‎via Consolare Valeria 1‎, ‎98125 MESSINA‎, ‎Italy

10.22072/wala.2017.52567.1091

Abstract

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integro-differential equation (FIDE) with a weakly singular kernel‎. ‎First‎, ‎a collocation method based on Haar wavelets (HW)‎, ‎Legendre wavelet (LW)‎, ‎Chebyshev wavelets (CHW)‎, ‎second kind Chebyshev wavelets (SKCHW)‎, ‎Cos and Sin wavelets (CASW) and BPFs are presented for driving approximate solution FIDEs with a weakly singular kernel‎. ‎Error estimates of all proposed numerical methods are given to test the convergence and accuracy of the method‎. ‎A comparative study of accuracy and computational time for the presented techniques is given‎.

Keywords


[1] S. Arbabi, A. Nazari and M.T. Darvishi, A two-dimensional Haar wavelets method for solving systems of PDEs. Appl. Math. Comput., 292 (2017), 33-46.
[2] C. Cattani, Local Fractional Calculus on Shannon Wavelet Basis, Nonlinear Physical Science, 2015.
[3] C. Cattani,  Shannon wavelets for the solution of integro-differential equations, Math. Probl. Eng., 2010 (2010), doi:10.1155/2010/408418.
[4] C. Cattani, Shannon wavelets theory, Math. Probl. Eng., 2008 (2008), doi:10.1155/2008/164808.
[5] C. Cattani and A. Kudreyko,  Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. Math. Comput., 215(12) (2010), 4164-4171.
[6] I. Celik, Chebyshev Wavelet collocation method for solving generalized Burgers–Huxley equation, Math. Methods Appl. Sci., 39(3) (2016), 366--377.
[7] C.K. Chui, An Introduction to Wavelets, (Wavelet Analysis and Its Applications), vol. 1, Elsevier Press, Amsterdam 1992.
[8] E. Hernandez and  G. Weiss, A First Course on Wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996.
[9] M.H. Heydari, M.R. Hooshmandasl, F.M. Ghaini and C. Cattani,  Wavelets method for solving fractional optimal control problems, Appl. Math. Comput., 286} (2016) 139-154.
[10] M.H. Heydari, M.R. Hooshmandasl and  F. Mohammadi,  Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation, Adv. Appl. Math. Mech., 6(2) (2014), 247-260.
[11] M.R. Hooshmandasl, M.H. Heydari and C. Cattani, Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions, Eur. Phys. J. Plus, (2016) 131-268, doi:10.1140/epjp/i2016-16268-2.
[12] A. Kilicman and Z.A. Zhour, Kronecker operational matrices for fractional  calculus and some applications, Appl. Math. Comput., 187(1) (2007),  250-65.
[13] Y. Li, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2284--2292.
[14] F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations, Int. J. Appl. Math. Res., 4(2) (2015), 217-227.
[15] F. Mohammadi,  A wavelet-based computational method for solving stochastic Ito-Volterra integral equations, J. Comput. Phys., 298 (2015), 254-265.
[16] F. Mohammadi, Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix of fractional derivative, Int. J. Adv. Appl. Math. Mech., 2(1) (2014), 83-91.
[17] F. Mohammadi, Numerical solution of stochastic Ito-Volterra integral equations using Haar wavelets, Numer. Math., Theory Methods Appl., 9(3) (2016), 416-431.
[18] F. Mohammadi and M.M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst., 348(8) (2011),  1787-1796.
[19] F. Mohammadi and  M.M. Hosseini, Legendre wavelet method for solving linear stiff systems, Journal of Advanced Research in Differential Equations, 2(1) (2010), 47-57.
[20] S.T. Mohyud-Din, A. Ali and B. Bin-Mohsin, On biological population model of fractional order, Int. J. Biomath., 9 (2016), doi:10.1142/S1793524516500704.
[21] S.T. Mohyud-Din, A. Waheed and M.M. Rashidi, A study of nonlinear age-structured population models, Int. J. Biomath., 9 (2016), doi:10.1142/S1793524516500911.
[22] I. Podlubny,  Fractional Differential Equations, Academic Press, San Diego, 1999.
[23] P. Rahimkhani,  Y. Ordokhani, and E.  Babolian, A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations, Numer. Algorithms, 74(1) (2017), 223-245.    
[24] S.S. Ray and A.K. Gupta, Numerical solution of fractional partial differential equation of parabolic type with Dirichlet boundary conditions using Two-dimensional Legendre wavelets method, Journal of Computational and Nonlinear Dynamics, 11(1) (2016), doi:10.1115/1.4028984.
[25] H. Saeedi,   Application of the haar wavelets in solving nonlinear fractional Fredholm intergro-differential equations, J. Mahani Math. Res. Cent., 2(1) (2012), 15-28.
[26] H. Saeedi and  M. Mohseni Moghadam, Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets, Commun. Nonlinear Sci. Numer. Simul., 16(3) (2011),  1216-1226.
[27] H. Saeedi, M. Mohseni Moghadam, M. Mollahasani and G.N. Chuev, A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order,  Commun. Nonlinear. Sci. Numer. Simul., 16(3) (2011), 1154-63.
[28] S.C. Shiralashetti and A. B. Deshi, An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations, Nonlinear Dyn., 83(2) (2016), 293-303.
[29] Y. Wang and Q. Fan, The second kind Chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput., 218(17) (2012), 8592-8601.
[30] Y. Wang and L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput., 275 (2016), 72-80.
[31] X.J. Yang, J.T. Machado, D. Baleanu and C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26(8) (2016), doi: 10.1063/1.4960543.
[32] X.J. Yang,  J.T. Machado and  J. Hristov,  Nonlinear dynamics for local fractional Burgers equation arising in fractal flow, Nonlinear Dyn., 84(1) (2016), 3-7.
[33] X.J. Yang, J.T. Machado and H.M. Srivastava,  A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Appl. Math. Comput., 274 (2016), 143-151.
[34] X.J. Yang, H.M. Srivastava and C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67(3) (2015), 752-761.