Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

Maysaa Al Qurashi; Saima Rashid; Sobia Sultana; Fahd Jarad; Abdullah M. Alsharif; Maysaa Al Qurashi; Saima Rashid; Sobia Sultana; Fahd Jarad; Abdullah M. Alsharif

doi:10.3934/math.2022697

AIMS Mathematics

2022, Volume 7, Issue 7: 12587-12619. doi: 10.3934/math.2022697

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Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

1.
Department of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh 12211, Saudi Arabia
4.
Department of Mathematics, Cankaya University, Ankara, Turkey
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
6.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
7.
Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 26 February 2022 Revised: 14 April 2022 Accepted: 19 April 2022 Published: 28 April 2022
MSC : 46S40, 47H10, 54H25

In this research, the $ \bar{\mathbf{q}} $-homotopy analysis transform method ($ \bar{\mathbf{q}} $-HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $ \bar{\mathbf{q}} $-HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.
Citation: Maysaa Al Qurashi, Saima Rashid, Sobia Sultana, Fahd Jarad, Abdullah M. Alsharif. Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory[J]. AIMS Mathematics, 2022, 7(7): 12587-12619. doi: 10.3934/math.2022697

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Abstract

In this research, the $ \bar{\mathbf{q}} $-homotopy analysis transform method ($ \bar{\mathbf{q}} $-HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $ \bar{\mathbf{q}} $-HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.

References

[1]	H. I. Abdel-Gawad, M. Tantawy, D. Baleanu, Fractional KdV and Boussenisq-Burger's equations, reduction to PDE and stability approaches, Math. Method. Appl. Sci., 43 (2020), 4125–4135. http://doi.org/10.1002/mma.6178 doi: 10.1002/mma.6178
[2]	H. I. Abdel-Gawad, M. Tantawy, Traveling wave solutions of DNA-Torsional model of fractional order, Appl. Math. Inf. Sci. Lett., 6 (2018), 85–89. http://doi.org/10.18576/amisl/060205 doi: 10.18576/amisl/060205
[3]	N. I. Okposo, P. Veeresha, E. N. Okposo, Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons, Chinese J. Phy., 77 (2022), 965–984. http://doi.org/10.1016/j.cjph.2021.10.014 doi: 10.1016/j.cjph.2021.10.014
[4]	P. Veeresha, M. Yavuz, C. Baishya, A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators, IJOCTA, 11 (2021), 52–67. http://doi.org/10.11121/ijocta.2021.1177 doi: 10.11121/ijocta.2021.1177
[5]	P. Veeresha, E. Ilhan, H. M. Baskonus, Fractional approach for analysis of the model describing wind-influenced projectile motion, Phy. Scr., 96 (2021), 075209. http://doi.org/10.1088/1402-4896/abf868 doi: 10.1088/1402-4896/abf868
[6]	P. Veeresha, D. Baleanu, A unifying computational framework for fractional Gross-Pitaevskii equations, Phy. Scr., 96 (2021), 125010. http://doi.org/10.1088/1402-4896/ac28c9 doi: 10.1088/1402-4896/ac28c9
[7]	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. http://doi.org/10.1063/1.5074099 doi: 10.1063/1.5074099
[8]	C. Baishya, P. Veeresha, Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel, Proc. R. Soc. A, 477 (2021), 20210438. http://doi.org/10.1098/rspa.2021.0438 doi: 10.1098/rspa.2021.0438
[9]	S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y.-M. Chu, On multi-step methods for singular fractional $q$-integro-differential equations, Open Math., 19 (2021), 1378–1405. http://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
[10]	F. Jin, Z.-S. Qian, Y.-M. Chu, M. ur Rahman, On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative, J. Appl. Anal. Comput., 12 (2022), 790–806. http://doi.org/10.11948/20210357 doi: 10.11948/20210357
[11]	S. Rashid, E. I. Abouelmagd, A. Khalid, F. B. Farooq, Y.-M. Chu, Some recent developments on dynamical $\hbar$-discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels, Fractals, 30 (2022), 2240110. http://doi.org/10.1142/S0218348X22401107 doi: 10.1142/S0218348X22401107
[12]	F.-Z. Wang, M. N. Khan, I. Ahmad, H. Ahmad, H. Abu-Zinadah, Y.-M. Chu, Numerical solution of traveling waves in chemical kinetics: time-fractional Fishers equations, Fractals, 30 (2022), 2240051. http://doi.org/10.1142/S0218348X22400515 doi: 10.1142/S0218348X22400515
[13]	S. Rashid, E. I. Abouelmagd, S. Sultana, Y.-M. Chu, New developments in weighted $n$-fold type inequalities via discrete generalized h-proportional fractional operators, Fractals, 30 (2022), 2240056. http://doi.org/10.1142/S0218348X22400564 doi: 10.1142/S0218348X22400564
[14]	Y.-M. Chu, S. Bashir, M. Ramzan, M. Y. Malik, Model-based comparative study of magnetohydrodynamics unsteady hybrid nanofluid flow between two infinite parallel plates with particle shape effects, Math. Method. Appl. Sci., in press. http://doi.org/10.1002/mma.8234
[15]	S. A. Iqbal, M. G. Hafez, Y.-M. Chu, C. Park, Dynamical Analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative, J. Appl. Anal. Comput., 12 (2022), 770–789. http://doi.org/10.11948/20210324 doi: 10.11948/20210324
[16]	C. Baishya1, S. J. Achar, P. Veeresha, D. G. Prakasha, Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos, 31 (2021), 043130. http://doi.org/10.1063/5.0028905 doi: 10.1063/5.0028905
[17]	E. Ilhan, P. Veeresha, H. M. Baskonus, Fractional approach for a mathematical model of atmospheric dynamics of CO2 gas with an efficient method, Chaos Soliton. Fract., 152 (2021), 111347. http://doi.org/10.1016/j.chaos.2021.111347 doi: 10.1016/j.chaos.2021.111347
[18]	P. Veeresha, H. M. Baskonus, W. Gao, Strong interacting internal waves in rotating ocean: Novel fractional approach, Axioms, 10 (2021), 123. http://doi.org/10.3390/axioms10020123 doi: 10.3390/axioms10020123
[19]	P. Veeresha, D. G. Prakasha, D. Kumar, An efficient technique for nonlinear time-fractional Klein–Fock–Gordon equation, Appl. Math. Comput., 364 (2020), 124637. http://doi.org/10.1016/j.amc.2019.124637 doi: 10.1016/j.amc.2019.124637
[20]	P. Veeresha, D. G. Prakasha, M. A. Qurashi, D. Baleanu, A reliable technique for fractional modified Boussinesq and approximate long wave equations, Adv. Differ. Equ., 2019 (2019), 253. http://doi.org/10.1186/s13662-019-2185-2 doi: 10.1186/s13662-019-2185-2
[21]	T.-H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math., 20 (2021), 160–176.
[22]	M. Nazeer, F. Hussain, M. Ijaz Khan, Asad-ur-Rehman, E. R. El-Zahar, Y.-M. Chu, Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel, Appl. Math. Comput., 420 (2022), 126868. http://doi.org/10.1016/j.amc.2021.126868 doi: 10.1016/j.amc.2021.126868
[23]	Y.-M. Chu, B. M. Shankaralingappa, B. J. Gireesha, F. Alzahrani, M. Ijaz Khan, S. U. Khan, Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano- material surface, Appl. Math. Comput., 419 (2022), 126883. http://doi.org/10.1016/j.amc.2021.126883 doi: 10.1016/j.amc.2021.126883
[24]	T.-H. Zhao, M. Ijaz Khan, Y.-M. Chu, Artificial neural networking (ANN) analysis for heat and entropy generation in flow of non-Newtonian fluid between two rotating disks, Math. Method. Appl. Sci., in press. http://doi.org/10.1002/mma.7310
[25]	K. Karthikeyan, P. Karthikeyan, H. M. Baskonus, K. Venkatachalam, Y.-M. Chu, Almost sectorial operators on $\Psi$-Hilfer derivative fractional impulsive integro-differential equations, Math. Method. Appl. Sci., in press. https://doi.org/10.1002/mma.7954
[26]	Y.-M. Chu, U. Nazir, M. Sohail, M. M. Selim, J.-R. Lee, Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach, Fractal Fract., 5 (2021), 119. http://doi.org/10.3390/fractalfract5030119 doi: 10.3390/fractalfract5030119
[27]	S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y.-M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), 2240026. http://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
[28]	B. S. T. Alkahtani, A. Atangana, Analysis of non-homogenous heat model with new trend of derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 566–571. https://doi.org/10.1016/j.chaos.2016.03.027 doi: 10.1016/j.chaos.2016.03.027
[29]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
[30]	T.-H. Zhao, M.-K. Wang, G.-J. Hai, Y.-M. Chu, Landen inequalities for Gaussian hypergeometric function, RACSAM, 116, (2022), 53. https://doi.org/10.1007/s13398-021-01197-y
[31]	T.-H. Zhao, M.-K. Wang, Y.-M. Chu, On the bounds of the perimeter of an ellipse, Acta Math. Sci., 42 (2022), 491–501. https://doi.org/10.1007/s10473-022-0204-y doi: 10.1007/s10473-022-0204-y
[32]	T.-H. Zhao, Z.-Y. He, Y.-M. Chu, Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals, Comput. Methods Funct. Theory, 21 (2021), 413–426. https://doi.org/10.1007/s40315-020-00352-7 doi: 10.1007/s40315-020-00352-7
[33]	L. Wang, X. Chen, Approximate analytical solutions of time fractional Whitham–Broer–Kaup equations by a residual power series method, Entropy, 17 (2015), 6519–6533. http://doi.org/10.3390/e17096519 doi: 10.3390/e17096519
[34]	S. Rashid, K. T. Kubra, H. Jafari, S. U. Lehre, A semi-analytical approach for fractional order Boussinesq equation in a gradient unconfined aquifers, Math. Method. Appl. Sci., 45 (2022), 1033–1062. http://doi.org/10.1002/mma.7833 doi: 10.1002/mma.7833
[35]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, Novel simulations to the time-fractional Fisher's equation, Math. Sci., 13 (2019), 33–42. http://doi.org/10.1007/s40096-019-0276-6 doi: 10.1007/s40096-019-0276-6
[36]	M. I. El-Bahi, K. Hilal, Lie symmetry analysis, exact solutions, and conservation laws for the generalized time-fractional KdV-Like equation, J. Funct. Space., 2021 (2021), 6628130. http://doi.org/10.1155/2021/6628130 doi: 10.1155/2021/6628130
[37]	S. C. Shiralashetti, S. Kumbinarasaiah, Laguerre wavelets collocation method for the numerical solution of the Benjamina–Bona-Mohany equations, J. Taibah Univ. Sci., 13 (2019), 9–15. http://doi.org/10.1080/16583655.2018.1515324 doi: 10.1080/16583655.2018.1515324
[38]	A. Kadem, D. Baleanu, On fractional coupled Whitham–Broer–Kaup equations, Rom. J. Phys., 56 (2011), 629–635.
[39]	G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A, 299 (1967), 6–25. http://doi.org/10.1098/rspa.1967.0119 doi: 10.1098/rspa.1967.0119
[40]	L. J. F. Broer, Approximate equations for long water waves, Appl. Sci. Res., 31 (1975), 377–395. http://doi.org/10.1007/BF00418048 doi: 10.1007/BF00418048
[41]	D. J. Kaup, A higher-order water-wave equation and the method for solving it, Progress of Theoretical Physics, 54 (1975), 396–408. http://doi.org/10.1143/PTP.54.396 doi: 10.1143/PTP.54.396
[42]	M. Al-Qurashi, S. Rashid, F. Jarad, M. Tahir, A. M. Alsharif, New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method, AIMS Mathematics, 7 (2022), 2044–2060. http://doi.org/10.3934/math.2022117 doi: 10.3934/math.2022117
[43]	S. Rashid, F. Jarad, T. M. Jawa, A study of behaviour for fractional order diabetes model via the nonsingular kernel, AIMS Mathematics, 7 (2022), 5072–5092. http://doi.org/10.3934/math.2022282 doi: 10.3934/math.2022282
[44]	S. Rashid, S. Sultana, R. Ashraf, M. K. A. Kaabar, On comparative analysis for the Black-Scholes model in the generalized fractional derivatives sense via Jafari transform, J. Funct. Space., 2021 (2021), 7767848. http://doi.org/10.1155/2021/7767848 doi: 10.1155/2021/7767848
[45]	S. T. Mohyud-Din, A. Ydrm, G. Demirli, Traveling wave solutions of Whitham–Broer–Kaup equations by homotopy perturbation method, J. King Saud Univ. Sci., 22 (2010), 173–176. http://doi.org/10.1016/j.jksus.2010.04.008 doi: 10.1016/j.jksus.2010.04.008
[46]	S. Saha Ray, A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water, Math. Method. Appl. Sci., 38 (2015), 1352–1368. http://doi.org/10.1002/mma.3151 doi: 10.1002/mma.3151
[47]	B. Tian, Y. Qiu, Exact and explicit solutions of Whitham–Broer–Kaup equations in shallow water, Pure and Applied Mathematics Journal, 5 (2016), 174–180. http://doi.org/10.11648/j.pamj.20160506.11 doi: 10.11648/j.pamj.20160506.11
[48]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, An efficient technique for coupled fractional Whitham–Broer–Kaup equations describing the propagation of shallow water waves, In: 4th International conference on computational mathematics and engineering sciences (CMES-2019), Cham: Springer, 2019, 49–75. http://doi.org/10.1007/978-3-030-39112-6_4
[49]	J. Singh, D. Kumar, R. Swroop, Numerical solution of time- and space-fractional coupled Burgers' equations via homotopy algorithm, Alex. Eng. J., 55 (2016), 1753–1763. http://doi.org/10.1016/j.aej.2016.03.028 doi: 10.1016/j.aej.2016.03.028
[50]	S. J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simul., 2 (1997), 95–100. https://doi.org/10.1016/S1007-5704(97)90047-2 doi: 10.1016/S1007-5704(97)90047-2
[51]	S. J. Liao, Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech., 19 (1998), 957–962. http://doi.org/10.1007/BF02457955 doi: 10.1007/BF02457955
[52]	K. S. Aboodh, The new integral transform "Aboodh transform", Global Journal of Pure and Applied Mathematics, 9 (2013), 35–43.
[53]	H. Bulut, D. Kumar, J. Singh, R. Swroop, H. M. Baskonus, Analytic study for a fractional model of HIV infection of CD4+T lymphocyte cells, Math. Nat. Sci., 2 (2018), 33–43. http://doi.org/10.22436/mns.02.01.04 doi: 10.22436/mns.02.01.04
[54]	P. Veeresha, D. G. Prakasha, Solution for fractional Zakharov-Kuznetsov equations by using two reliable techniques, Chinese J. Phys., 60 (2019), 313–330. http://doi.org/10.1016/j.cjph.2019.05.009 doi: 10.1016/j.cjph.2019.05.009
[55]	D. G. Prakasha, P. Veeresha, Analysis of lakes pollution model with Mittag-Leffler kernel, J. Ocean Eng. Sci., 5 (2020), 310–322. http://doi.org/10.1016/j.joes.2020.01.004 doi: 10.1016/j.joes.2020.01.004
[56]	M. H. Cherif, D. Ziane, A new numerical technique for solving systems of nonlinear fractional partial differential equations, Int. J. Anal. Appl., 15 (2017), 188–197.
[57]	M. A. Awuya, D. Subasi, Aboodh transform iterative method for solving fractional partial differential equation with Mittag–Leffler kernel, Symmetry, 13 (2021), 2055. http://doi.org/10.3390/sym13112055 doi: 10.3390/sym13112055
[58]	G. M. Mittag-Leffler, Sur La nonvelle Fonction $E_{\alpha}(x)$, C. R. Acad. Sci. Paris, (Ser. II), 137 (1903), 554–558.
[59]	S. M. El-Sayed, D. Kaya, Exact and numerical travelling wave solutions of Whitham–Broer–Kaup equations, Appl. Math. Comput., 167 (2005), 1339–1349. http://doi.org/10.1016/j.amc.2004.08.012 doi: 10.1016/j.amc.2004.08.012
[60]	M. Rafei, H. Daniali, Application of the variational iteration method to the Whitham-Broer-Kaup equations, Comput. Math. Appl, 54 (2007), 1079–1085. http://doi.org/10.1016/j.camwa.2006.12.054 doi: 10.1016/j.camwa.2006.12.054

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