NEW OPERATIONAL MATRIX OF RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OF ORTHONORMAL BERNOULLI POLYNOMIALS FOR THE NUMERICAL SOLUTION OF SOME DISTRIBUTED-ORDER TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

M. Pourbabaee; A. Saadatmandi

doi:10.11948/20230039

2023 Volume 13 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(6): 3352-3373

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M. Pourbabaee, A. Saadatmandi. NEW OPERATIONAL MATRIX OF RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OF ORTHONORMAL BERNOULLI POLYNOMIALS FOR THE NUMERICAL SOLUTION OF SOME DISTRIBUTED-ORDER TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3352-3373. doi: 10.11948/20230039

Citation:

M. Pourbabaee, A. Saadatmandi. NEW OPERATIONAL MATRIX OF RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OF ORTHONORMAL BERNOULLI POLYNOMIALS FOR THE NUMERICAL SOLUTION OF SOME DISTRIBUTED-ORDER TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3352-3373. doi: 10.11948/20230039

NEW OPERATIONAL MATRIX OF RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OF ORTHONORMAL BERNOULLI POLYNOMIALS FOR THE NUMERICAL SOLUTION OF SOME DISTRIBUTED-ORDER TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

M. Pourbabaee^1, ,,
A. Saadatmandi^1,

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran

Author Bio: Email: a.saadatmandi@gmail.com(A. Saadatmandi)
Corresponding author: Email: m.pourbabaee@kashanu.ac.ir or mr.pourbabaee@gmail.com(M. Pourbabaee)

Abstract

Abstract

In this article, the orthonormal Bernoulli polynomials (OBPs) and their properties are applied for concluding a general technique for forming a new operational matrix of the distributed-order (DO) fractional derivative. Then, we apply tau approach and obtained operational matrix to solve some DO time-fractional partial differential equations including distributed-order Rayleigh-Stokes problem (DRSP) for a generalized second-grade fluid and DO anomalous sub-diffusion equation. Our methodology reduces the solution of these problems to a set of algebraic equations. By analysis the error of approximation by the obtained matrix and comparing between the numerical solutions and exact result, we can conclude that this operational matrix is valid to solve the mentioned equations. Also, to confirm the accuracy and the validity of our technique three examples are provided. Finally, we compare obtained results from this approach with the achieved results from relevant studies.
- Distributed-order /
- Riemann-Liouville derivative /
- orthonormal Bernoulli polynomials /
- fractional differential equations /
- Rayleigh-Stokes problem
MSC: 26A33, 65M70

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References

[1]	M. Abbaszadeh and M. Dehghan, An improved meshless method for solving two-dimensional distributed-order time fractional diffusion-wave equation with error estimate, Numer. Algor., 2017, 75, 173–211. doi: 10.1007/s11075-016-0201-0 CrossRef Google Scholar
[2]	M. Abbaszadeh and M. Dehghan, Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation, Appl. Math. Comput., 2021, 392, 125718. Google Scholar
[3]	M. A. Abdelkawy and R. T. Alqahtani, Shifted Jacobi collocation method for solving multi-dimensional fractional Stokes' first problem for a heated generalized second grade fluid, Adv. Difference Equ., 2016, 114, 17pp. Google Scholar
[4]	A. A. Alikhanov, Numerical methods of solutions of boundary value problems for the multi-term variable-distributed-order diffusion equation, Appl. Math. Comput., 2015, 268, 12–22. Google Scholar
[5]	S. Bazm, Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations, J. Comput. Appl. Math., 2015, 275, 44–60. doi: 10.1016/j.cam.2014.07.018 CrossRef Google Scholar
[6]	A. H. Bhrawy, M. A. Zaky and J. F. Alzaidy, Two shifted Jacobi-Gauss collocation schemes for solving two-dimensional variable-order fractional Rayleigh-Stokes problem, Adv. Difference Equ., 2016, 272, 17pp. Google Scholar
[7]	W. Bu, A. Xiao and W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 2017, 72, 422–441. doi: 10.1007/s10915-017-0360-8 CrossRef Google Scholar
[8]	M. Caputo, Linear models of dissipation whose q is almost frequency independent -ii, Geophys J. R. Astr. Soc., 1967, 13, 529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x CrossRef Google Scholar
[9]	Y. Chen and C. M. Chen, Numerical algorithm for solving the Stokes' first problem for a heated generalized second grade fluid with fractional derivative, Numer. Algorithms, 2018, 77, 939–953. doi: 10.1007/s11075-017-0348-3 CrossRef Google Scholar
[10]	H. Dehestani, Y. Ordokhani and M. Razzaghi, Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations, Appl. Math. Comput., 2018, 336, 433–453. Google Scholar
[11]	M. Dehghan and M. Abbaszadeh, A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Eng. Comput., 2017, 33, 587–605. doi: 10.1007/s00366-016-0491-9 CrossRef Google Scholar
[12]	K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math., 2009, 225, 96–104. doi: 10.1016/j.cam.2008.07.018 CrossRef Google Scholar
[13]	W. Fan and F. Liu, A numerical method for solving the two-dimensional distributed-order space-fractional diffusion equation on an irregular convex domain, Appl. Math. Lett., 2018, 77, 114–121. doi: 10.1016/j.aml.2017.10.005 CrossRef Google Scholar
[14]	G. Fernández-Anaya, G. Nava-Antonio, J. Jamous-Galante, R. Muñoz-Vega and E. G. Hernández-Martinez, Asymptotic stability of distributed-order nonlinear dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 2017, 48, 541–549. doi: 10.1016/j.cnsns.2017.01.020 CrossRef Google Scholar
[15]	N. J. Ford, M. L. Morgado and M. Rebelo, An implicit finite difference approximation for the solution of the diffusion equation with distributed-order in time, Electron. Trans. Numer. Anal., 2015, 44, 289–305. Google Scholar
[16]	G. Gao, A. A. Alikhanov and Z. Sun, The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations, J. Sci. Comput., 2017, 73, 93–121. doi: 10.1007/s10915-017-0407-x CrossRef Google Scholar
[17]	G. H. Gao, H. W. Sun and Z. Z. Sun, Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys., 2015, 298, 337–359. doi: 10.1016/j.jcp.2015.05.047 CrossRef Google Scholar
[18]	G. H. Gao and Z. Z. Sun, Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations, Comput. Math. Appl., 2015, 69, 926–948. Google Scholar
[19]	A. Golbabai and S. Panjeh Ali Beik, An efficient method based on operational matrices of Bernoulli polynomials for solving matrix differential equations, Comput. Appl. Math., 2015, 34, 159–175. doi: 10.1007/s40314-013-0110-y CrossRef Google Scholar
[20]	R. M. Hafez, M. A. Zaky and M. A. Abdelkawy, Jacobi spectral galerkin method for distributed-order fractional Rayleigh-Stokes problem for a generalized second grade fluid, Front. Phys., 2020, 7, 1–10. Google Scholar
[21]	A. Haghbin and H. Jafari, Solving time-fractional chemical engineering equations by modified variational iteration method as fixed point iteration method, Iranian J. Math. Chem., 2017, 8, 365–375. Google Scholar
[22]	M. H. Heydari and Z. Avazzadeh, New formulation of the orthonormal Bernoulli polynomials for solving the variable-order time fractional coupled Boussinesq-Burger's equations, Eng Comput., 2021, 37, 3509–3517. doi: 10.1007/s00366-020-01007-w CrossRef Google Scholar
[23]	M. H. Heydari, M. Hosseininia and D. Baleanu, A computational approach based on the fractional Euler functions and Chebyshev cardinal functions for distributed-order time fractional 2D diffusion equation, Alex. Eng. J., 2023, 67, 643–653. doi: 10.1016/j.aej.2022.12.065 CrossRef Google Scholar
[24]	M. H. Heydari, S. Rashid and Y. -M. Chu, Chelyshkov polynomials method for distributed-order time fractional nonlinear diffusion-wave equations, Results Phys., 2023, 47, 106344. doi: 10.1016/j.rinp.2023.106344 CrossRef Google Scholar
[25]	M. H. Heydari, S. Rashid and F. Jarad, A numerical method for distributed-order time fractional 2D Sobolev equation, Results Phys., 2023, 45, 106211. doi: 10.1016/j.rinp.2023.106211 CrossRef Google Scholar
[26]	M. H. Heydari, M. Razzaghi and D. Baleanu, Numerical solution of distributed-order time fractional Klein–Gordon–Zakharov system, J. Comput. Sci., 2023, 67, 101961. doi: 10.1016/j.jocs.2023.101961 CrossRef Google Scholar
[27]	M. H. Heydari, M. Razzaghi and D. Baleanu, A numerical method based on the piecewise Jacobi functions for distributed-order fractional Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 2023, 116, 106873. doi: 10.1016/j.cnsns.2022.106873 CrossRef Google Scholar
[28]	F. B. Hildebrand, Introduction to Numerical Analysis, Dover Publications, New York, 1956. Google Scholar
[29]	Z. Jiao, Y. Chen and I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer, New York, 2012. Google Scholar
[30]	E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978. Google Scholar
[31]	C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, New York, 2015. Google Scholar
[32]	L. Li, F. Liu, L. Feng and I. Turner, A Galerkin finite element method for the modified distributed-order anomalous sub-diffusion equation, J. Comput. Appl. Math., 2020, 368, 112589. doi: 10.1016/j.cam.2019.112589 CrossRef Google Scholar
[33]	Y. Lin and W. Jiang, Numerical method for Stokes' first problem for a heated generalized second grade fluid with fractional derivative, Numer. Methods Partial Differential Equations, 2011, 27, 1599–1609. doi: 10.1002/num.20598 CrossRef Google Scholar
[34]	S. Mashayekhi, Y. Ordokhani and M. Razzaghi, Hybrid functions approach for nonlinear constrained optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 1831–1843. doi: 10.1016/j.cnsns.2011.09.008 CrossRef Google Scholar
[35]	S. Mashayekhi and M. Razzaghi, Numerical solution of distributed-order fractional differential equations by hybrid functions, J. Comput. Phys., 2016, 315, 169–181. doi: 10.1016/j.jcp.2016.01.041 CrossRef Google Scholar
[36]	A. Mohebbi, M. Abbaszadeh and M. Dehghan, Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives, Comput. Methods Appl. Mech. Engrg., 2013, 264, 163–177. doi: 10.1016/j.cma.2013.05.012 CrossRef Google Scholar
[37]	M. L. Morgado and M. Rebelo, Numerical approximation of distributed-order reaction-diffusion equations, J. Comput. Appl. Math., 2015, 275, 216–227. doi: 10.1016/j.cam.2014.07.029 CrossRef Google Scholar
[38]	M. L. Morgado, M. Rebelo, L. L. Ferrás and N. J. Ford, Numerical solution for diffusion equations with distributed-order in time using a Chebyshev collocation method, Appl. Numer. Math., 2017, 114, 108–123. doi: 10.1016/j.apnum.2016.11.001 CrossRef Google Scholar
[39]	V. G. Pimenov, A. S. Hendy and R. H. De Staelen, On a class of non-linear delay distributed-order fractional diffusion equations, J. Comput. Appl. Math., 2017, 318, 433–443. doi: 10.1016/j.cam.2016.02.039 CrossRef Google Scholar
[40]	I. Podulbny, Fractional Differential Equations, Academic Press, New York, 1999. Google Scholar
[41]	M. Pourbabaee and A. Saadatmandi, A novel Legendre operational matrix for distributed-order fractional differential equations, Appl. Math. Comput., 2019, 361, 215–231. Google Scholar
[42]	M. Pourbabaee and A. Saadatmandi, A new operational matrix based on Müntz-Legendre polynomials for solving distributed-order fractional differential equations, Math. Comput. Simulation, 2022, 194, 210–235. doi: 10.1016/j.matcom.2021.11.023 CrossRef Google Scholar
[43]	P. Rahimkhani, Y. Ordokhani and E. Babolian, Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations, Appl. Numer. Math., 2017, 122, 66-81. doi: 10.1016/j.apnum.2017.08.002 CrossRef Google Scholar
[44]	T. J. Rivlin, An Introduction to the Approximation of Functions, Dover Publications, New York, 1981. Google Scholar
[45]	A. Saadatmandi, A. Khani and M. R. Azizi, Numerical calculation of fractional derivatives for the Sinc functions via Legendre polynomials, Math. Interdisc. Res., 2020, 5, 71–86. Google Scholar
[46]	N. Samadyar and F. Mirzaee, Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itô-Volterra integral equations of Abel type, Int. J. Numer. Model., 2019, 33, e2688. Google Scholar
[47]	F. Shen, W. C. Tan, Y. Zhao and T. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlinear Anal. Real World Appl., 2006, 7, 1072–1080. doi: 10.1016/j.nonrwa.2005.09.007 CrossRef Google Scholar
[48]	E. Shivanian and A. Jafarabadi, Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation, Eng Comput., 2018, 34, 77–90. doi: 10.1007/s00366-017-0522-1 CrossRef Google Scholar
[49]	U. P. Singh, Application of orthonormal Bernoulli polynomials for approximate solution of some Volterra integral equations, Albanian J. Math., 2016, 10, 47–80. Google Scholar
[50]	C. Wu, Numerical solution for Stokes' first problem for a heated generalized second grade fluid with fractional derivative, Appl. Numer. Math., 2009, 59, 2571–2583. doi: 10.1016/j.apnum.2009.05.009 CrossRef Google Scholar
[51]	X. Yang and X. Jiang, Numerical algorithm for two dimensional fractional Stokes' first problem for a heated generalized second grade fluid with smooth and non-smooth solution, Comput. Math. Appl., 2019, 78, 1562–1571. Google Scholar
[52]	C. Ye, X. N. Luo and L. P. Wen, High-order numerical methods of fractional-order Stokes' first problem for heated generalized second grade fluid, Appl. Math. Mech. (English Ed.), 2012, 33, 65–80. doi: 10.1007/s10483-012-1534-8 CrossRef Google Scholar
[53]	H. Ye, F. Liu and V. Anh, Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, J. Comput. Phys., 2015, 298, 652–660. doi: 10.1016/j.jcp.2015.06.025 CrossRef Google Scholar
[54]	H. Ye, F. Liu, V. Anh and I. Turner, Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math., 2015, 80, 825–838. doi: 10.1093/imamat/hxu015 CrossRef Google Scholar
[55]	M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 2018, 75, 2243–2258. Google Scholar