Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative

Xiaoyong Xu; Fengying Zhou; Xiaoyong Xu; Fengying Zhou

doi:10.3934/math.2023144

AIMS Mathematics

2023, Volume 8, Issue 2: 2736-2762. doi: 10.3934/math.2023144

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Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative

Xiaoyong Xu ^,,
Fengying Zhou

School of Science, East China University of Technology, Jiangxi, Nanchang 330013, China

Received: 10 August 2022 Revised: 27 September 2022 Accepted: 01 November 2022 Published: 09 November 2022
MSC : 34A08, 34K28, 65T60

In this paper, a new orthonormal wavelets based on the orthonormal Euler polynomials (OEPs) is constructed to approximate the numerical solution of time-fractional Cattaneo equation with Caputo-Fabrizio derivative. By applying the Gram-Schmidt orthonormalization process on sets of Euler polynomials of various degrees, an explicit representation of OEPs is obtained. The convergence analysis and error estimate of the orthonormal Euler wavelets expansion are studied. The exact formula of Caputo-Fabrizio fractional integral of orthonormal Euler wavelets are derived using Laplace transform. The applicability and validity of the proposed method are verified by some numerical examples.
- orthonormal Euler wavelets,
- Caputo-Fabrizio fractional integral,
- Cattaneo equation,
- Laplace transform,
- convergence analysis
Citation: Xiaoyong Xu, Fengying Zhou. Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative[J]. AIMS Mathematics, 2023, 8(2): 2736-2762. doi: 10.3934/math.2023144

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Abstract

In this paper, a new orthonormal wavelets based on the orthonormal Euler polynomials (OEPs) is constructed to approximate the numerical solution of time-fractional Cattaneo equation with Caputo-Fabrizio derivative. By applying the Gram-Schmidt orthonormalization process on sets of Euler polynomials of various degrees, an explicit representation of OEPs is obtained. The convergence analysis and error estimate of the orthonormal Euler wavelets expansion are studied. The exact formula of Caputo-Fabrizio fractional integral of orthonormal Euler wavelets are derived using Laplace transform. The applicability and validity of the proposed method are verified by some numerical examples.

References

[1]	N. H. Tuan, H. Mohammadi, S. Rezapour, A mathematical model for COVID-19 transmission by using the Caputo fractional derivative, Chaos Solitons Fract., 140 (2020), 110107. https://doi.org/10.1016/j.chaos.2020.110107 doi: 10.1016/j.chaos.2020.110107
[2]	S. Kumar, R. Kumar, J. Singh, K. S. Nisar, D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of CD4$ ^{+} $ T-cells with the effect of antiviral drug therapy, Alex. Eng. J., 59 (2020), 2053–2064. https://doi.org/10.1016/j.aej.2019.12.046 doi: 10.1016/j.aej.2019.12.046
[3]	R. Almeida, A. M. C. B. da Cruz, N. Martins, M. T. T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, Int. J. Dyn. Control, 7 (2019), 776–784. https://doi.org/10.1007/s40435-018-0492-1 doi: 10.1007/s40435-018-0492-1
[4]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29 (2019), 013119. https://doi.org/10.1063/1.5074099 doi: 10.1063/1.5074099
[5]	R. Ali, M. I. Asjad, A. Aldalbahi, M. Rahimi-Gorji, M. Rahaman, Convective flow of a Maxwell hybrid nanofluid due to pressure gradient in a channel, J. Therm. Anal. Calorim., 143 (2021), 1319–1329. https://doi.org/10.1007/s10973-020-10304-x doi: 10.1007/s10973-020-10304-x
[6]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[7]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Solitons Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[8]	A. Atangana, B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: application of Caputo-Fabrizio derivative, Arab. J. Geosci. 9 (2016), 1–6. https://doi.org/10.1007/s12517-015-2060-8 doi: 10.1007/s12517-015-2060-8
[9]	A. Atangana, D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017), D4016005. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001091 doi: 10.1061/(ASCE)EM.1943-7889.0001091
[10]	A. Alshabanat, M. Jleli, S. Kumar, B. Samet, Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits, Front. Phys., 8 (2020), 64. https://doi.org/doi:10.3389/fphy.2020.00064 doi: 10.3389/fphy.2020.00064
[11]	M. ur Rahman, S. Ahmad, R. T. Matoog, N. A. Alshehri, T. Khan, Study on the mathematical modelling of COVID-19 with Caputo-Fabrizio operator, Chaos Solitons Fract., 150 (2021), 111121. https://doi.org/10.1016/j.chaos.2021.111121 doi: 10.1016/j.chaos.2021.111121
[12]	E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Math. Model. Anal., 21 (2016), 188–198. https://doi.org/10.3846/13926292.2016.1145607 doi: 10.3846/13926292.2016.1145607
[13]	F. Gao, X. J. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Therm. Sci., 20 (2016), 871–877. https://doi.org/10.2298/TSCI16S3871G doi: 10.2298/TSCI16S3871G
[14]	O. J. Peter, A. Yusuf, K. Oshinubi, F. A. Oguntolu, J. O. Lawal, A. I. Abioye, et al., Fractional order of pneumococcal pneumonia infection model with Caputo Fabrizio operator, Results Phys., 29 (2021), 104581. https://doi.org/10.1016/j.rinp.2021.104581 doi: 10.1016/j.rinp.2021.104581
[15]	O. J. Peter, Transmission dynamics of fractional order brucellosis model using Caputo-Fabrizio operator, Int. J. Differ. Equ., 2020 (2020), 2791380. https://doi.org/10.1155/2020/2791380 doi: 10.1155/2020/2791380
[16]	E. Awad, On the time-fractional Cattaneo equation of distributed order, Phys. A, 518 (2019), 210–233. https://doi.org/10.1016/j.physa.2018.12.005 doi: 10.1016/j.physa.2018.12.005
[17]	A. Compte, R. Metzler, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A Math. Gen., 30 (1997), 7277.
[18]	K. D. Lewandowska, T. Kosztolowicz, Application of generalized Cattaneo equation to model subdiffusion impedance, Acta Phys. Pol. B, 39 (2008), 1211–1220.
[19]	H. R Ghazizadeh, M. Maerefat, A. Azimi, Explicit and implicit finite difference schemes for fractional Cattaneo equation, J. Comput. Phys., 229 (2010), 7042–7057. https://doi.org/10.1016/j.jcp.2010.05.039 doi: 10.1016/j.jcp.2010.05.039
[20]	S. W. Vong, H. K. Pang, X. Q. Jin, A high-order difference scheme for the generalized Cattaneo equation, East Asian J. Appl. Math., 2 (2012), 170–184. https://doi.org/10.4208/eajam.110312.240412a doi: 10.4208/eajam.110312.240412a
[21]	X. Zhao, Z. Z. Sun, Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium, J. Sci. Comput., 62 (2015), 747–771. https://doi.org/10.1007/s10915-014-9874-5 doi: 10.1007/s10915-014-9874-5
[22]	J. C. Ren, G. H. Gao, Efficient and stable numerical methods for the two-dimensional fractional Cattaneo equation, Numer. Algorithms, 69 (2015), 795–818. https://doi.org/10.1007/s11075-014-9926-9 doi: 10.1007/s11075-014-9926-9
[23]	L. L. Wei, Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation, Numer. Algorithms, 77 (2018), 675–690. https://doi.org/10.1007/s11075-017-0334-9 doi: 10.1007/s11075-017-0334-9
[24]	Y. M. Wang, A Crank-Nicolson-type compact difference method and its extrapolation for time fractional Cattaneo convection-diffusion equations with smooth solutions, Numer. Algorithms, 81 (2019), 489–527. https://doi.org/10.1007/s11075-018-0558-3 doi: 10.1007/s11075-018-0558-3
[25]	L. J. Nong, Q. Yi, J. X. Cao, A. Chen, Fast compact difference scheme for solving the two-dimensional time-fractional Cattaneo equation, Fractal Fract., 6 (2022), 438. https://doi.org/10.3390/fractalfract6080438 doi: 10.3390/fractalfract6080438
[26]	Z. G. Liu, A. J. Cheng, X. L. Li, A second order Crank-Nicolson scheme for fractional Cattaneo equation based on new fractional derivative, Appl. Math. Comput., 311 (2017), 361–374. https://doi.org/10.1016/j.amc.2017.05.032 doi: 10.1016/j.amc.2017.05.032
[27]	M. Taghipour, H. Aminikhah, A $\theta$-finite difference scheme based on cubic B-spline quasi-interpolation for the time fractional Cattaneo equation with Caputo-Fabrizio operator, J. Differ. Equ. Appl., 27 (2021), 712–738. https://doi.org/10.1080/10236198.2021.1935909 doi: 10.1080/10236198.2021.1935909
[28]	M. Yaseen, Q. U. Nisa Arif, R. George, S. Khan, Comparative numerical study of spline-based numerical techniques for time fractional Cattaneo equation in the sense of Caputo-Fabrizio, Fractal Fract., 6 (2022), 50. https://doi.org/10.3390/fractalfract6020050 doi: 10.3390/fractalfract6020050
[29]	H. N. Li, S. J. Lü, T. Xu, A fully discrete spectral method for fractional Cattaneo equation based on Caputo-Fabrizo derivative, Numer. Methods Partial Differ. Equ., 35 (2019), 936–954. https://doi.org/10.1002/num.22332 doi: 10.1002/num.22332
[30]	E. Hesameddini, M. Shahbazi, Two-dimensional shifted Legendre polynomials operational matrix method for solving the two-dimensional integral equations of fractional order, Appl. Math. Comput., 322 (2018), 40–54. https://doi.org10.1016/j.amc.2017.11.024 doi: 10.1016/j.amc.2017.11.024
[31]	H. Singh, C. S. Singh, A reliable method based on second kind Chebyshev polynomial for the fractional model of Bloch equation, Alex. Eng. J., 57 (2018), 1425–1432. https://doi.org/10.1016/j.aej.2017.07.002 doi: 10.1016/j.aej.2017.07.002
[32]	S. Jaiswal, S. Das, Numerical solution of linear/nonlinear fractional order differential equations using Jacobi operational matrix, Int. J. Appl. Comput. Math., 5 (2019), 1–21. https://doi.org/10.1007/s40819-019-0625-z doi: 10.1007/s40819-019-0625-z
[33]	A. Isah, C. Phang, New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials, J. King Saud Univ. Sci., 31 (2019), 1–7. https://doi.org/10.1016/j.jksus.2017.02.001 doi: 10.1016/j.jksus.2017.02.001
[34]	S. Sadeghi Roshan, H. Jafari, D. Baleanu, Solving FDEs with Caputo-Fabrizio derivative by operational matrix based on Genocchi polynomials, Math. Methods Appl. Sci., 41 (2018), 9134–9141. https://doi.org/10.1002/mma.5098 doi: 10.1002/mma.5098
[35]	J. R. Loh, A. Isah, C. Phang, Y. T. Toh, On the new properties of Caputo-Fabrizio operator and its application in deriving shifted Legendre operational matrix, Appl. Numer. Math., 132 (2018), 138–153. https://doi.org/10.1016/j.apnum.2018.05.016 doi: 10.1016/j.apnum.2018.05.016
[36]	S. Kumar, J. F. Gómez Aguilar, P. Pandey, Numerical solutions for the reaction-diffusion, diffusion-wave, and Cattaneo equations using a new operational matrix for the Caputo-Fabrizio derivative, Math. Methods Appl. Sci., 43 (2020), 8595–8607. https://doi.org/10.1002/mma.6517 doi: 10.1002/mma.6517
[37]	B. Yuttanan, M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Model., 70 (2019), 350–364. https://doi.org/10.1016/j.apm.2019.01.013 doi: 10.1016/j.apm.2019.01.013
[38]	N. Kumar, M. Mehra, Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost, Numer. Methods Partial Differ. Equ., 37 (2021), 1693–1724. https://doi.org/10.1002/num.22604 doi: 10.1002/num.22604
[39]	G. Esra Köse, Ö. Oruç A. Esen, An application of Chebyshev wavelet method for the nonlinear time fractional Schröinger equation, Math. Methods Appl. Sci., 45 (2022), 6635–6649. https://doi.org/10.1002/mma.8196 doi: 10.1002/mma.8196
[40]	J. Shahni, R. Singh, A fast numerical algorithm based on Chebyshev-wavelet technique for solving Thomas-Fermi type equation, Eng. Comput., 2021, 1–14. https://doi.org/10.1007/s00366-021-01476-7 doi: 10.1007/s00366-021-01476-7
[41]	A. Secer, M. Cinar, A Jacobi wavelet collocation method for fractional Fisher's equation in time, Therm. Sci., 24 (2020), 119–129. https://doi.org/10.2298/tsci20s1119s doi: 10.2298/tsci20s1119s
[42]	M. R. Eslahchi, M. Kavoosi, The use of Jacobi wavelets for constrained approximation of rational Béier curves, Comput. Appl. Math., 37 (2018), 3951–3966. https://doi.org/10.1007/s40314-017-0552-8 doi: 10.1007/s40314-017-0552-8
[43]	A. T. Turan Dincel, Solution to fractional-order Riccati differential equations using Euler wavelet method, Sci. Iran., 26 (2019), 1608–1616. https://doi.org/10.24200/SCI.2018.51246.2084 doi: 10.24200/SCI.2018.51246.2084
[44]	S. Behera, S. Saha Ray, Euler wavelets method for solving fractional-order linear Volterra-Fredholm integro-differential equations with weakly singular kernels, Comput. Appl. Math., 40 (2021), 1–30. https://doi.org/10.1007/s40314-021-01565-9 doi: 10.1007/s40314-021-01565-9
[45]	R. Aruldoss, R. A. Devi, P. M. Krishna, An expeditious wavelet-based numerical scheme for solving fractional differential equations, Comput. Appl. Math., 40 (2021), 1–14. https://doi.org/10.1007/s40314-020-01387-1 doi: 10.1007/s40314-020-01387-1
[46]	J. L. Schiff, The Laplace transform: theory and applications, Undergraduate Texts in Mathematics, New York: Springer, 1999.
[47]	M. Moumen Bekkouche, H. Guebbai, M. Kurulay, S. Benmahmoud, A new fractional integral associated with the Caputo-Fabrizio fractional derivatives, Rend. Circ. Mat. Palermo (2), 70 (2021), 1277–1288. https://doi.org/10.1007/s12215-020-00557-8 doi: 10.1007/s12215-020-00557-8
[48]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[49]	Y. He, Some new results on products of Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 431 (2015), 34–46. https://doi.org/10.1016/j.jmaa.2015.05.055 doi: 10.1016/j.jmaa.2015.05.055
[50]	F. Mohammadi, A. Ciancio, Wavelet-based numerical method for solving fractional integro-differential equation with a weakly singular kernel, Wavelet Linear Algebra, 4 (2017), 53–73. https://doi.org/10.22072/wala.2017.52567.1091 doi: 10.22072/wala.2017.52567.1091
[51]	M. M. Bekkouche, H. Guebbai, M. Kurulay, On the solvability fractional of a boundary value problem with new fractional integral, J. Appl. Math. Comput., 64 (2020), 551–564. https://doi.org/10.1007/s12190-020-01368-x doi: 10.1007/s12190-020-01368-x

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