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Investigation of fuzzy fractional Kuramoto–Sivashinsky equations by an efficient approach

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Abstract

In this article, we examine the Kuramoto–Sivashinsky equation, a nonlinear model that modifies a variety of physical and chemical scenarios. The main goal of this article is to find the analytical solution to the fuzzy fractional Kuramoto–Sivashinsky equations (FFKSEs). We employ the fractional reduced differential transform method (FRDTM) for dealing with the fractional Caputo operator. By using three examples, we tested the developed technique. The upper and lower parts of the fuzzy solutions for all three challenges were obtained using two different fractional-order simulations between 0 and 1. The attained series-type result was identified in the current investigation. The results show that the suggested strategy leads to accurate solutions at the integer level. The outcomes of applying the suggested methodology demonstrate the effectiveness and simplicity of our approach for analysing the behaviour of nonlinear models used in science and technology.

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References

  • Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360(1), 109–113 (2006)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  • Ahmad, J., Mohyud-Din, S.T.: An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics. PLoS ONE 9(12), e109127 (2014)

  • Ahmad, J., Iqbal, A., Hassan, Q.M.U.: Study of nonlinear fuzzy integro-differential equations using mathematical methods and applications. Int J Fuzzy Logic Intell Syst 21(1), 76–85 (2021)

    Article  Google Scholar 

  • Akgül, A.: A novel method for a fractional derivative with non-local and non-singular kernel. Chaos Solitons Fractals 114, 478–482 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  • Akgül, A., Cordero, A., Torregrosa, J.R.: A fractional Newton method with 2ath-order of convergence and its stability. Appl. Math. Lett. 98, 344–351 (2019)

    Article  MathSciNet  Google Scholar 

  • Al-Ghafri, K.S., Alabdala, A.T., Redhwan, S.S., Bazighifan, O., Ali, A.H., Iambor, L.F.: Symmetrical solutions for non-Local fractional integro-differential equations via Caputo–Katugampola derivatives. Symmetry 15(3), 662 (2023)

  • Ali, A., Islam, S., Khan, M.R., Rasheed, S., Allehiany, F.M., Baili, J., Ahmad, H.: Dynamics of a fractional order Zika virus model with mutant. Alexandria Eng. J. 61(6), 4821–4836 (2022)

    Article  Google Scholar 

  • Ali, A., Alshammari, F.S., Islam, S., Khan, M.A., Ullah, S.: Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative. Results Phys. 20, 103669 (2021)

  • Aljahdaly, N. H., Naeem, M., Wyal, N.: Analysis of fuzzy Kuramoto–SSivashinsky equations under a generalized fuzzy fractional derivative operator. J. Funct. Spaces 2022, 1–11 (2022)

  • Arafa, A.A.M., Khalil, M., Sayed, A.: A non-integer variable order mathematical model of human immunodeficiency virus and malaria coinfection with time delay. Complexity 2019, 1–13 (2019)

  • Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020)

  • Behzadi, S.S., Allahviranloo, T., Abbasbandy, S.: Solving fuzzy second-order nonlinear Volterra–Fredholm integro-differential equations by using Picard method. Neural Comput. Appl. 21, 337–346 (2012)

    Article  Google Scholar 

  • Boutarfa, B., Akgül, A., Inc, M.: New approach for the Fornberg–Whitham type equations. J. Comput. Appl. Math. 312, 13–26 (2017)

    Article  MathSciNet  Google Scholar 

  • Caputo, M.: Elasticita de dissipazione, Zanichelli, Bologna (1969)

  • Chakraverty, S., Jena, R.M., Jena, S.K.: Time-Fractional Order Biological Systems with Uncertain Parameters, vol. 12, pp. 1–160. Morgan, Claypool Publishers, San Rafael (2011)

  • Choudhary, R., Kumar, D.: Numerical solution of linear time-fractional Kuramoto–Sivashinsky equation via quintic B-splines. Int. J. Comput. Math. 100(7), 1512–1531 (2023)

  • de Oliveira, E.C., Mainardi, F., Vaz, J., Jr.: Fractional models of anomalous relaxation based on the Kilbas and Saigo function. Meccanica 49(9), 2049–2060 (2014)

    Article  MathSciNet  Google Scholar 

  • Dhaigude, D.B., Bhadgaonkar, V.N.: A novel approach for fractional Kawahara and modified Kawahara equations using Atangana–Baleanu derivative operator. J. Math. Comput. Sci. 11(3), 2792–2813 (2021)

    Google Scholar 

  • Dhaigude, D.B., Kiwne, S.B., Dhaigude, R.M.: Monotone iterative scheme for weakly coupled system of finite difference reaction–diffusion equations. Communications in Applied Analysis 12(2), 161 (2008)

  • Dubois, D., Prade, H.: Towards fuzzy differential calculus part 1: integration of fuzzy mappings. Fuzzy Sets Syst. 8(1), 1–17 (1982)

    Article  Google Scholar 

  • El-Saka, H.A.A., Arafa, A.A.M., Gouda, M.I.: Dynamical analysis of a fractional SIRS model on homogenous networks. Adv. Differ. Equ. 2019(1), 1–15 (2019)

    Article  MathSciNet  Google Scholar 

  • Figueiredo Camargo, R., Capelas de Oliveira, E., Vaz, J., Jr.: On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator. J. Math. Phys. 50(12), 123518 (2009)

  • Fishelov, D., Croisille, J.P.: Optimal convergence for time-dependent linearized Kuramoto–Sivashinsky type problems: a new approach. J. Comput. Appl. Math. 429, 115229 (2023)

  • Gaul, L., Klein, P., Kemple, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5(2), 81–88 (1991)

    Article  ADS  Google Scholar 

  • Goetschel, R., Jr., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18(1), 31–43 (1986)

    Article  MathSciNet  Google Scholar 

  • Haq, E.U., Hassan, Q.M.U., Ahmad, J., Ehsan, K.: Fuzzy solution of system of fuzzy fractional problems using a reliable method. Alex. Eng. J. 61(4), 3051–3058 (2022)

    Article  Google Scholar 

  • He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  • Iqbal, A., Ahmad, J., Hassan, Q.M.U.: Application of an effective method on the system of nonlinear fuzzy integro-differential equations. J. Sci. Arts 21(2), 407–422 (2021)

    Article  Google Scholar 

  • Iqbal, N., Khan, I., Shah, R., Nonlaopon, K.: The fuzzy fractional acoustic waves model in terms of the Caputo-Fabrizio operator. AIMS Math. 8(1), 1770–1783 (2023)

    Article  MathSciNet  Google Scholar 

  • Jafari, M., Alipour Fakhri, Y., Khadivar, M.: Densities and fluxes of the conservation laws for the Kuramoto–Sivashinsky equation. J. Linear Topol. Algebra 11(01), 47–54 (2022)

    Google Scholar 

  • Jena, R.M., Chakraverty, S., Jena, S.K.: Analysis of the dynamics of phytoplankton nutrient and whooping cough models with nonsingular kernel arising in the biological system. Chaos Solitons Fractals 141, 110373 (2020)

  • Jena, R.M., Chakraverty, S., Jena, S.K., Sedighi, H.M.: Analysis of time-fractional fuzzy vibration equation of large membranes using double parametric based Residual power series method. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 101(5), e202000165 (2021)

  • Jena, R.M., Chakraverty, S., Yavuz, M., Abdeljawad, T.: A new modeling and existence-uniqueness analysis for Babesiosis disease of fractional order. Mod. Phys. Lett. B 35(30), 2150443 (2021)

  • Kaleva, O.: Fuzzy differential equations. Fuzzy Sets Syst. 24(3), 301–317 (1987)

    Article  MathSciNet  Google Scholar 

  • Kaur, A., Kanwar, V.: Numerical solution of generalized Kuramoto–Sivashinsky equation using cubic trigonometric B-spline based differential quadrature method and one-step optimized hybrid block method. Int. J. Appl. Comput. Math. 8, 1–19 (2022)

    Article  MathSciNet  Google Scholar 

  • Keskin, Y., Oturanc, G.: The reduced differential transform method: a new approach to fractional partial differential equations. Nonlinear Sci. Lett. A 1(2), 207–217 (2010)

    Google Scholar 

  • Khan, M.A., Ullah, S., Kumar, S.: A robust study on 2019-nCOV outbreaks through non-singular derivative. Eur. Phys. J. Plus 136, 1–20 (2021)

    Article  Google Scholar 

  • Kiliç, S. S. S., Çelik, E.: Complex solutions to the higher-order nonlinear Boussinesq type wave equation transform. Ricerche di Matematica 1–8 (2022)

  • Kumar, S., Chauhan, R.P., Momani, S., Hadid, S.: Numerical investigations on COVID-19 model through singular and non-singular fractional operators. Numer. Methods Partial Differ. Equ. 36(1), 1–27 (2020)

  • Kumar, D., Singh, J., Tanwar, K., Baleanu, D.: A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws. Int. J. Heat Mass Transf. 138, 1222–1227 (2019)

    Article  CAS  Google Scholar 

  • Kumar, S., Kumar, A., Samet, B., Dutta, H.: A study on fractional host-parasitoid population dynamical model to describe insect species. Numer. Methods Partial Differ. Equ. 37(2), 1673–1692 (2021)

    Article  MathSciNet  Google Scholar 

  • Kumar, S., Kumar, R., Osman, M.S., Samet, B.: A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials. Numer. Methods Partial Differ. Equ. 37(2), 1250–1268 (2021)

    Article  MathSciNet  Google Scholar 

  • Liu, P., Din, A., Zarin, R.: Numerical dynamics and fractional modeling of hepatitis B virus model with non-singular and non-local kernels. Results Phys. 39, 105757 (2022)

  • Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021)

  • Niazi, A.U.K., Iqbal, N., Shah, R., Wannalookkhee, F., Nonlaopon, K.: Controllability for fuzzy fractional evolution equations in credibility space. Fractal Fract. 5(3), 112 (2021)

  • Osman, M., Xia, Y., Omer, O.A., Hamoud, A.: On the fuzzy solution of linear-nonlinear partial differential equations. Mathematics 10(13), 2295 (2022)

  • Rahman, M.U., Arfan, M., Deebani, W., Kumam, P., Shah, Z.: Analysis of time-fractional Kawahara equation under Mittag–Leffler power law. Fractals 30(01), 2240021 (2022)

  • Ramani, P., Khan, A.M., Suthar, D.L., Kumar, D.: Approximate analytical solution for non-linear Fitzhugh–Nagumo equation of time fractional order through fractional reduced differential transform method. Int. J. Appl. Comput. Math. 8(2), 61 (2022)

  • Saad Alshehry, A., Imran, M., Khan, A., Shah, R., Weera, W.: Fractional view analysis of Kuramoto–Sivashinsky equations with non-singular kernel operators. Symmetry 14(7), 1463 (2022)

    Article  ADS  Google Scholar 

  • Seadawy, A.R., Iqbal, M., Lu, D.: Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg–de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A 544, 123560 (2020)

  • Shah, K., Seadawy, A.R., Arfan, M.: Evaluation of one dimensional fuzzy fractional partial differential equations. Alex. Eng. J. 59(5), 3347–3353 (2020)

    Article  Google Scholar 

  • Singh, J., Hristov, J.Y., Hammouch, Z. (eds.): New Trends in Fractional Differential Equations with Real-World Applications in Physics. Frontiers Media SA, Lausanne (2020)

  • 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)

    Article  Google Scholar 

  • Sitthiwirattham, T., Arfan, M., Shah, K., Zeb, A., Djilali, S., Chasreechai, S.: Semi-analytical solutions for fuzzy Caputo–Fabrizio fractional-order two-dimensional heat equation. Fractal Fract. 5(4), 139 (2021)

  • Sontakke, B.R., Shelke, A.S., Shaikh, A.S.: Solution of non-linear fractional differential equations by variational iteration method and applications. Far East J. Math. Sci. 110(1), 113–129 (2019)

    Google Scholar 

  • Tazgan, T., Celik, E., Gülnur, Y.E.L., Bulut, H.: On survey of the some wave solutions of the non-linear Schrödinger equation (NLSE) in Infinite Water Depth. Gazi Univ. J. Sci. 36(2), 819–843 (2023)

  • Thangavelu, Padmasekaran, S.: The exact solutions of heat equation by RDTM for the fractional order. In: AIP Conference Proceedings 2261, 1–5 (2020)

  • ur Rahman, M., Arfan, M., Shah, Z., Alzahrani, E.: Evolution of fractional mathematical model for drinking under Atangana-Baleanu Caputo derivatives. Physica Scripta 96(11), 115203 (2021)

  • Wu, P., Din, A., Munir, T., Malik, M.Y., Alqahtani, A.S.: Local and global Hopf bifurcation analysis of an age-infection HIV dynamics model with cell-to-cell transmission. Waves Random Complex Media 1–16 (2022)

  • Yazgan, T., Ilhan, E., Çelik, E., Bulut, H.: On the new hyperbolic wave solutions to Wu–Zhang system models. Opt. Quant. Electron. 54(5), 298 (2022)

  • Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  Google Scholar 

  • Zimmermann, H.J.: Fuzzy Set Theory and Its Applications. Springer, Berlin (2011)

    Google Scholar 

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  1. Department of mathematics, Faculty of Science, University of Gujrat, Gujrat, 50700, Pakistan

    Jamshad Ahmad & Fatima Nusrat

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  1. Jamshad Ahmad
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  2. Fatima Nusrat
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JA: Resources, supervision, validation, acquisition. FN: Conceptualization, methodology, writing—original draft, formal analysis, software.

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Ahmad, J., Nusrat, F. Investigation of fuzzy fractional Kuramoto–Sivashinsky equations by an efficient approach. Opt Quant Electron 56, 53 (2024). https://doi.org/10.1007/s11082-023-05380-w

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