Abstract.
This study deals with a new class of fractional partial integro-differential equations (FPI-DEs) characterized by the presence of weakly singular kernel and a Newtonian viscoelasticity factor. To numerically solve such equations, a hybrid method is established by combining the Legendre wavelets (LWs), the collocation method, and a new operational matrix of fractional integration (OMFI). More precisely, the unknown solution is expanded by the LWs with unknown coefficients. Then, the OMFI and the collocation method are utilized to extract a system of algebraic equations whose solution is an approximation for the problem’s solution. Convergence and error estimation of the LWs expansion in two dimensions are investigated. Moreover, the efficiency and accuracy of the proposed method are demonstrated by solving some concrete examples. The obtained results confirm the presented approach is very accurate to provide satisfactory solutions.
Similar content being viewed by others
References
M.E. Gurtin, A.C. Pipkin, Arch. Ration. Mech. Anal. 31, 113 (1968)
R.K. Miller, J. Math. Anal. Appl. 66, 313 (1978)
R.M. Christensen, Theory of Viscoelasticity (Academic Press, New York, 1982) p. 378
M. Rcnardy, Annu. Rev. Fluid Mech. 21, 21 (1989)
T. Tang, Appl. Numer. Math. 11, 309 (1993)
K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)
M. Hajipour, A. Jajarmi, D. Baleanu, H.G. Sun, Commun. Nonlinear Sci. Numer. Simul. 69, 119 (2019)
R. Meng, D. Yin, S. Drapaca, Comput. Mech. (2019) https://doi.org/10.1007/s00466-018-1663-9
D. Baleanu, A. Jajarmi, M. Hajipour, Nonlinear Dyn. 94, 397 (2018)
A. Jajarmi, D. Baleanu, Chaos, Solitons Fractals 113, 221 (2018)
D. Baleanu, A. Jajarmi, E. Bonyah, M. Hajipour, Adv. Differ. Equ. 2018, 230 (2018)
A. Jajarmi, D. Baleanu, J. Vib. Control 24, 2430 (2018)
E. Hesameddini, A. Rahimi, E. Asadollahifard, Commun. Nonlinear Sci. Numer. Simul. 34, 154 (2016)
M. Chen, L. Jia, X. Chen, X. Yin, J. Sound. Vib. 333, 7183 (2014)
A. Calderon, B. Vinagre, Signal Process. 86, 2803 (2006)
Y.W.T. Li, Y. Yang, Discr. Dyn. Nat. Soc. 63, 1 (2014)
M. Khader, S. Kumar, Math. Method Appl. Sci. 37, 2972 (2014)
Z.H. Guo, O. Acan, S. Kumar, Therm. Sci. 20, 739 (2016)
Y. Zhang, A. Kumar, S. Kumar, D. Baleanu, X.J. Yang, J. Nonlinear Sci. Appl. 9, 5821 (2016)
S. Kumar, A. Kumar, I.K. Argyros, Numer. Algor. 75, 213 (2017)
A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010)
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 36, 4931 (2012)
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Comput. Math. Appl. 62, 2364 (2011)
S. Kazem, Abbasbandy, S. Kumar, Appl. Math. Model. 37, 5498 (2013)
A. Ahmadian, M. Suleiman, S. Salahshour, Abstr. Appl. Anal. 2013, 505903 (2013)
D. Baleanu, A.H. Bhrawy, T.M. Taha, Abstr. Appl. Anal. 2013, 546502 (2013)
M. Ishteva, L. Boyadjiev, C. R. Acad. Bulg. Sci. 58, 1019 (2005)
M. Ishteva, L. Boyadjiev, R. Scherer, Math. Sci. Res. 9, 161 (2005)
M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, Comput. Math. Appl. 68, 269 (2014)
A.H. Bhrawy, M.A. Zaky, R.A. Van Gorder, Numer. Algor. 71, 151 (2016)
A.H. Bhrawy, Numer. Algor. 73, 91 (2016)
A.H. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, M.A. Abdelkawy, Calcolo 53, 17 (2016)
C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, Berlin, 1988)
C.K. Chui, Wavelet Analysis and Its Applications (Academic Press, San Diego, 1992)
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, C. Cattani, Commun. Nonlinear Sci. Numer. Simul. 19, 37 (2014)
M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, F. Mohammadi, J. Appl. Math. 2012, 542401 (2012)
M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, M. Li, Adv. Math. Phys. 2013, 482083 (2013)
Y. Li, Commun. Nonlinear Sci. Numer. Simul. 15, 2284 (2010)
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Appl. Math. Comput. 234, 267 (2014)
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Adv. Appl. Math. Mech. 6, 247 (2014)
M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, C. Cattani, Phys. Lett. A 379, 71 (2015)
M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, C. Cattani, Appl. Math. Comput. 286, 139 (2016)
L. Zhu, Q. Fan, Commun. Nonlinear Sci. Numer. Simul. 17, 2333 (2012)
Y. Wang, Q. Fan, Appl. Math. Comput. 218, 8592 (2012)
A.K. Gupta, S.S. Ray, Appl. Math. Model. 39, 5121 (2015)
Y. Li, W.W. Zhao, Appl. Math. Comput. 216, 2276 (2010)
H. Saeedi, Commun. Nonlinear Sci. Numer. Simul. 16, 1154 (2011)
M.H. Heydari, J. Franklin Inst. 355, 4970 (2018)
M.H. Heydari, Z. Avazzadeh, M. Farzi Haromi, Appl. Math. Comput. 341, 215 (2019)
M.H. Heydari, Z. Avazzadeh, Chaos, Solitons Fractals 112, 180 (2018)
M.H. Heydari, Z. Avazzadeh, Asian J. Control 20, 1 (2018)
M. Hosseininia, M.H. Heydari, F.M. Maalek Ghaini, Z. Avazzadeh, Int. J. Nonlinear Sci. Numer. Simul. 19, 793 (2018)
M.H. Heydari, Z. Avazzadeh, Comput. Appl. Math. 37, 4397 (2018)
M.H. Heydari, M.R. Hooshmandasl, C. Cattani, G. Hariharan, Fund. Inform. 153, 173 (2017)
Ibrahim Çelik, Math. Methods Appl. Sci. 39, 366 (2016)
M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, F. Feriedouni, Eng. Anal. Bound. Elem. 37, 1331 (2013)
M.H. Heydari, J. Comput. Nonlinear Dyn. 11, 061014 (2016)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
M. Dehghan, Int. J. Comput. Math. 83, 123 (2006)
H. Chen, D. Xu, Numer. Math. Theor. Methods Appl. 5, 559 (2012)
M. William, M. Kassem, Numer. Math. 105, 481 (2007)
A.H. Bhrawya, M.A. Zakyc, J. Comput. Phys. 281, 876 (2015)
M. Gasca, T. Sauer, J. Comput. Appl. Math. 122, 23 (2000)
J. de Villiers, Mathematics of Approximation (Atlantis Press, 2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Avazzadeh, Z., Heydari, M.H. & Cattani, C. Legendre wavelets for fractional partial integro-differential viscoelastic equations with weakly singular kernels⋆. Eur. Phys. J. Plus 134, 368 (2019). https://doi.org/10.1140/epjp/i2019-12743-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2019-12743-6