Abstract
In this research, we analyze the non-linear time-fractional Kudryashov–Sinelshchikov equation, which represents waves induced by pressure inside the combination of gas bubbles and liquid, considering heat transfer and viscidness of liquid among gas bubbles and liquid. The non-linear Kudryashov–Sinelshchikov equation having fractional order is numerically solved employing the fractional reduced differential transform method (FRDTM). We have conducted a convergence analysis of the solution series obtained through FRDTM. Also, the FRDTM generates the solution without perturbation, discretization, or linearization. The numerical solution and error analysis of this non-linear Kudryashov–Sinelshchikov equation having fractional order with respect to time using FRDTM completely conforms with the exact solution as shown precisely in 2D and 3D graphs. The results of this research demonstrate the effectiveness and accuracy of the FRDTM in solving the non-linear time-fractional Kudryashov–Sinelshchikov equation.
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Kudryashov, N.A., Sinelshchikov, D.I.: Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. Sect. A Gen. At. Solid State Phys. 374(19–20), 2011–2016 (2010). https://doi.org/10.1016/j.physleta.2010.02.067
Gupta, A.K., Saha Ray, S.: On the solitary wave solution of fractional Kudryashov–Sinelshchikov equation describing nonlinear wave processes in a liquid containing gas bubbles. Appl. Math. Comput. 298, 1–12 (2017). https://doi.org/10.1016/j.amc.2016.11.003
Kumar, S., Niwas, M., Dhiman, S.K.: Abundant analytical soliton solutions and different wave profiles to the Kudryashov–Sinelshchikov equation in mathematical physics. J. Ocean Eng. Sci. 7(6), 565–577 (2022). https://doi.org/10.1016/j.joes.2021.10.009
Khalid, K.A., Maneea, M.: New approximation solution for time-fractional Kudryashov–Sinelshchikov equation using novel technique. Alex. Eng. J. 72, 559–572 (2023). https://doi.org/10.1016/j.aej.2023.04.027
Li, X., Tang, Y.: Interpolated coefficient mixed finite elements for semilinear time fractional diffusion equations. Fractal Fract. 7(6), 482 (2023). https://doi.org/10.3390/fractalfract7060482
Zhang, J., Zhang, X., Yang, B.: An approximation scheme for the time fractional convection-diffusion equation. Appl. Math. Comput. 335, 305–312 (2018)
Alwehebi, F., Hobiny, A., Maturi, D.: Variational iteration method for solving time fractional burgers equation using Maple. Appl. Math. 14(05), 336–348 (2023). https://doi.org/10.4236/am.2023.145021
Wang, H., Xu, X., Dou, J., Zhang, T., Wei, L.: Local discontinuous Galerkin method for the time-fractional KdV equation with the Caputo-Fabrizio fractional derivative. J. Appl. Math. Phys. 10(06), 1918–1935 (2022). https://doi.org/10.4236/jamp.2022.106132
Khan, A., Akram, T., Khan, A., Ahmad, S., Nonlaopon, K.: Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with nonsingular kernels. AIMS Math. 8(1), 1251–1268 (2023). https://doi.org/10.3934/math.2023063
Prajapati, V.J., Meher, R.: Solution of time-fractional Rosenau–Hyman model using a robust homotopy approach via formable transform. Iran. J. Sci. Technol. Trans. A Sci. 46(5), 1431–1444 (2022). https://doi.org/10.1007/s40995-022-01347-w
Okrasińska-Płociniczak, H., Płociniczak, Ł: Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line. Appl. Math. Comput. 424, 127033 (2022). https://doi.org/10.1016/j.amc.2022.127033
Mukhtar, S.: Numerical analysis of the time-fractional Boussinesq equation in gradient unconfined aquifers with the Mittag–Leffler derivative. Symmetry (Basel) 15(3), 608 (2023). https://doi.org/10.3390/sym15030608
Zhou, J.K.: Differential Transformation and Its Applications for Electronic Circuits, Huazhong Science & Technology University Press, China (1986). http://scholar.google.com.secure.sci-hub.io/scholar?q=J K Zhou Differential Transformation and Its Applications for Electrical Circuits Huazhong University Press Wuhan China 1986#1
Keskin, Y., Oturanç, G.: Reduced differential transform method for partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 10(6), 741–749 (2009). https://doi.org/10.1515/IJNSNS.2009.10.6.741
Gupta, P.K.: Approximate analytical solutions of fractional BenneyLin equation by reduced differential transform method and the homotopy perturbation method. Comput. Math. Appl. 61(9), 2829–2842 (2011). https://doi.org/10.1016/j.camwa.2011.03.057
Singh, B.K., Kumar, P.: FRDTM for numerical simulation of multi-dimensional, time-fractional model of Navier–Stokes equation. Ain Shams Eng. J. 9(4), 827–834 (2018). https://doi.org/10.1016/j.asej.2016.04.009
Tamboli, V.K., Tandel, P.V.: Solution of the time-fractional generalized Burger-Fisher equation using the fractional reduced differential transform method. J. Ocean Eng. Sci. 7(4), 399–407 (2022). https://doi.org/10.1016/j.joes.2021.09.009
Akinyemi, L., Akpan, U., Veeresha, P., Rezazadeh, H., Mustafa Inc: Computational techniques to study the dynamics of generalized unstable nonlinear Schrödinger equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.02.011
Kaplan, M., Akbulut, A.: The analysis of the soliton-type solutions of conformable equations by using generalized Kudryashov method. Opt. Quantum Electron. 53(9), 498 (2021). https://doi.org/10.1007/s11082-021-03144-y
Kaplan, M., Akbulut, A.: A mathematical analysis of a model involving an integrable equation for wave packet envelope. J. Math. (2022). https://doi.org/10.1155/2022/3486780
Raza, N., Rafiq, M.H., Kaplan, M., Kumar, S., Chu, Y.M.: The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations. Results Phys. 22, 103979 (2021). https://doi.org/10.1016/j.rinp.2021.103979
Raza, N., Murtaza, I.G., Sial, S., Younis, M.: On solitons: the biomolecular nonlinear transmission line models with constant and time variable coefficients. Waves Random Complex Media 28(3), 553–569 (2018). https://doi.org/10.1080/17455030.2017.1368734
Zubair, A., Raza, N., Mirzazadeh, M., Liu, W., Zhou, Q.: Analytic study on optical solitons in parity-time-symmetric mixed linear and nonlinear modulation lattices with non-Kerr nonlinearities. Optik (Stuttg). 173, 249–262 (2018). https://doi.org/10.1016/j.ijleo.2018.08.023
Raza, N., Zubair, A.: Optical dark and singular solitons of generalized nonlinear Schrödinger’s equation with anti-cubic law of nonlinearity. Mod. Phys. Lett. B 33(13), 1950158 (2019). https://doi.org/10.1142/S0217984919501586
Raza, N., Javid, A.: Dynamics of optical solitons with Radhakrishnan–Kundu–Lakshmanan model via two reliable integration schemes. Optik (Stuttg) 178, 557–566 (2019). https://doi.org/10.1016/j.ijleo.2018.09.133
Khan, K.A., Butt, A.R., Raza, N., Maqbool, K.: Unsteady magneto-hydrodynamics flow between two orthogonal moving porous plates. Eur. Phys. J. Plus 134(1), 1 (2019). https://doi.org/10.1140/epjp/i2019-12286-x
Raza, N., Arshed, S., Javid, A.: Optical solitons and stability analysis for the generalized second-order nonlinear Schrödinger equation in an optical fiber. Int. J. Nonlinear Sci. Numer. Simul. 21(7–8), 855–863 (2020). https://doi.org/10.1515/ijnsns-2019-0287
Raza, N., Seadawy, A.R., Kaplan, M., Butt, A.R.: Symbolic computation and sensitivity analysis of nonlinear Kudryashov’s dynamical equation with applications. Phys. Scr. 96(10), 105216 (2021). https://doi.org/10.1088/1402-4896/ac0f93
Ali, K.K., Maneea, M.: Optical soliton solutions for space fractional Schrödinger equation using similarity method. Results Phys. 46, 106284 (2023). https://doi.org/10.1016/j.rinp.2023.106284
Khalid, K.A., Maneea, M.: Optical solitons using optimal homotopy analysis method for time-fractional (1+1)-dimensional coupled nonlinear Schrodinger equations. Optik (Stuttg). 283, 170907 (2023). https://doi.org/10.1016/j.ijleo.2023.170907
Fan, Z.Y., Ali, K.K., Maneea, M., Yao, S.W., Mustafa Inc: Solution of time fractional Fitzhugh–Nagumo equation using semi analytical techniques. Results Phys. 51, 106679 (2023). https://doi.org/10.1016/j.rinp.2023.106679
Ali, K.K., Maneea, M., Mohamed, M.S.: Solving nonlinear fractional models in superconductivity using the q-homotopy analysis transform method. J. Math. (2023). https://doi.org/10.1155/2023/6647375
Sene, N.: Analytical solutions of a class of fluids models with the Caputo fractional derivative. Fractal Fract. 6(1), 35 (2022). https://doi.org/10.3390/fractalfract6010035
Lan, K.: Linear first order Riemann–Liouville fractional differential and perturbed Abel’s integral equations. J. Differ. Equ. 306, 28–59 (2022). https://doi.org/10.1016/j.jde.2021.10.025
Singh, B.K., Srivastava, V.K.: Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM. R. Soc. Open Sci. 2(4), 140511 (2015). https://doi.org/10.1098/rsos.140511
Srivastava, V.K., Kumar, S., Awasthi, M.K., Singh, B.K.: Two-dimensional time fractional-order biological population model and its analytical solution. Egypt. J. Basic Appl. Sci. 1(1), 71–76 (2014). https://doi.org/10.1016/j.ejbas.2014.03.001
Keskin, Y., Oturanç, G.: Reduced differential transform method for generalized KdV equations. Math. Comput. Appl. 15(3), 382–393 (2010). https://doi.org/10.3390/mca15030382
Abbasbandy, S.: Numerical method for non-linear wave and diffusion equations by the variational iteration method. Int. J. Numer. Methods Eng. 73(12), 1836–1843 (2008). https://doi.org/10.1002/nme.2150
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Tamboli, V.K., Tandel, P.V. Solution of the non-linear time-fractional Kudryashov–Sinelshchikov equation using fractional reduced differential transform method. Bol. Soc. Mat. Mex. 30, 24 (2024). https://doi.org/10.1007/s40590-024-00602-x
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DOI: https://doi.org/10.1007/s40590-024-00602-x
Keywords
- Fractional reduced differential transform method (FRDTM)
- Non-linear fractional Kudryashov–Sinelshchikov equation
- Non-linear fractional partial differential equation
- Caputo fractional derivative