Application of Fixed Point Theory and Solitary Wave Solutions for the Time-Fractional Nonlinear Unsteady Convection-Diffusion System
Tóm tắt
In this article, the two-dimensional time fractional unsteady convection-diffusion system is under consideration. The convection-diffusion system of nonlinear partial differential equations has remained a uniform fascination for scientists owing to its energetic significance as well as its possession of a broad spectrum of practical and physical applications. In particular, these practicable implications include turbulence, heat transfer, fluid flow, traffic flow, and modeling of gas dynamics. The Caputo operator is applied for the fractional order derivatives and their inversion. The existence of results and uniqueness is proved by applying the fixed point theory with the help of some well-known results and theorems such as the contraction mapping theorem with Lipschitz condition, and Schauder’s fixed point theorem. Mainly, we find the exact solitary wave solutions of the underlying model. For this sake, the new extended direct algebraic method is applied and the solutions are gained in the form of dark, singular, complex, combo, trigonometric and rational solutions. Further, we draw 3D plots to show the behavior of these solutions by choosing the different values of parameters.
Tài liệu tham khảo
Sabatier, J. A. T. M. J., Agrawal, O. P., Machado, J. T.: Advances in fractional calculus (Vol. 4, No. 9). Springer, Dordrecht (2007)
Das, S.: Functional fractional calculus (Vol. 1). Springer, Berlin (2011)
Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4(1), 75–89 (1977)
Jiang, X., Xu, M., Qi, H.:The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes. Nonlinear Anal.: Real World Appl. 11(1), 262–269 (2010)
Sene, N.: Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion model. Int. J. Anal. Appl. 17(2), 191–207 (2019)
Kumar, N.: Supercloseness analysis of a stabilizer free weak Galerkin finite element method for time dependent convection diffusion reaction equation. Math. Comput. Simul. 208, 582–602 (2023)
Atangana, A., Secer, A.: A note on fractional order derivatives and table of fractional derivatives of some special functions. In Abstract and applied analysis (Vol. 2013). Hindawi (2013)
Saqib, M., Hasnain, S., Mashat, D.S.: Highly efficient computational methods for two dimensional coupled nonlinear unsteady convection-diffusion problems. IEEE Access 5, 7139–7148 (2017)
Hasan, Z.A., Ali, A.S. J.: The Comparison Study of the Hybrid Method for Solving the Unsteady State Two-Dimensional Convection-Diffusion Equations. J. Adv. Res. Fluid Mech. Therm. Sci. 99(2), 67–86 (2022)
Zhao, Y., Huang, M., Ouyang, X., Luo, J., Shen, Y., Bao, F.: A half boundary method for two dimensional unsteady convection-diffusion equations. Eng. Anal. Bound. Elem. 135, 322–336 (2022)
Sheu, T.W., Chen, C.F., Hsieh, L.W.: Development of a sixth-order two-dimensional convection-diffusion scheme via Cole-Hopf transformation. Comput. Methods Appl. Mech. Eng. 191(27–28), 2979–2995 (2002)
Al-Saif, A. S. J., Hasan, Z. A.: An analytical approximate method for solving unsteady state two-dimensional convection-diffusion equations. J. Adv. Math. 21, 73–88 (2022)
Ahmad, Z., Bonanomi, G., di Serafino, D., Giannino, F.: Transmission dynamics and sensitivity analysis of pine wilt disease with asymptomatic carriers via fractal-fractional differential operator of Mittag-Leffler kernel. Appl. Numer. Math. 185, 446–465 (2023)
Ahmad, Z., El-Kafrawy, S. A., Alandijany, T. A., Giannino, F., Mirza, A. A., El-Daly, M. M., ... Azhar, E. I.: A global report on the dynamics of COVID-19 with quarantine and hospitalization: A fractional order model with non-local kernel. Comput. Biol. Chem. 98, 107645 (2022)
Khan, N., Ahmad, Z., Ahmad, H., Tchier, F., Zhang, X. Z., Murtaza, S.: Dynamics of chaotic system based on image encryption through fractal-fractional operator of non-local kernel. AIP Adv. 12(5), (2022)
Murtaza, S., Ahmad, Z., Ali, I.E., Akhtar, Z., Tchier, F., Ahmad, H., Yao, S.W.: Analysis and numerical simulation of fractal-fractional order non-linear couple stress nanofluid with cadmium telluride nanoparticles. J. King Saud University-Sci. 35(4), 102618 (2023)
Khan, N., Ahmad, Z., Shah, J., Murtaza, S., Albalwi, M. D., Ahmad, H., ... Yao, S. W.: Dynamics of chaotic system based on circuit design with Ulam stability through fractal-fractional derivative with power law kernel. Sci. Rep. 13(1), 5043 (2023)
Ahmad, Z., Ali, F., Khan, N., Khan, I.: Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel. Chaos, Solitons, Fractals 153, 111602 (2021)
Ngondiep, E.: A novel three-level time-split approach for solving two-dimensional nonlinear unsteady convection-diffusion-reaction equation. J. Math. Comput. Sci. 26(3), 222–248 (2022)
Chang, S. S.: Fixed point theory and application (1984)
Agarwal, R. P., O’Regan, D., Sahu, D. R.: Fixed point theory for Lipschitzian-type mappings with applications (Vol. 6, pp. x+-368). Springer, New York (2009)
Anley, E.F., Basha, M., Hussain, A., Dai, B.: Numerical simulation for nonlinear space-fractional reaction convection-diffusion equation with its application. Alex. Eng. J. 65, 245–261 (2023)
Ignat, L.I., Rossi, J.D.: A nonlocal convection-diffusion equation. J. Funct. Anal. 251(2), 399–437 (2007)
Haque, M.: Existence of weak solutions to a convection-diffusion equation in amalgam spaces. J. Egypt. Math. Soc. 30(1), 1–19 (2022)
Abbasbandy, S., Kazem, S., Alhuthali, M.S., Alsulami, H.H.: Application of the operational matrix of fractional-order Legendre functions for solving the time-fractional convection-diffusion equation. Appl. Math. Comput. 266, 31–40 (2015)
Gill, W.N., Sankarasubramanian, R.: Exact analysis of unsteady convective diffusion. Proc. Roy. Soc. London. A. Math. Phys. Sci. 316(1526), 341–350 (1970)
Kalita, J.C., Dalal, D.C., Dass, A.K.: A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients. Int. J. Numer. Methods Fluids 38(12), 1111–1131 (2002)
Ghayad, M.S., Badra, N.M., Ahmed, H.M., Rabie, W.B.: Derivation of optical solitons and other solutions for nonlinear Schrödinger equation using modified extended direct algebraic method. Alex. Eng. J. 64, 801–811 (2023)
Rehman, H.U., Ullah, N., Asjad, M.I., Akgül, A.: Exact solutions of convective-diffusive Cahn-Hilliard equation using extended direct algebraic method. Numer. Methods Partial Diff, Eq (2020)
Iqbal, M. S., Ahmed, N., Akgül, A., Raza, A., Shahzad, M., Iqbal, Z., ... Jarad, F.: Analysis of the fractional diarrhea model with Mittag-Leffler kernel. AIMS Math 7, 13000–13018 (2022)
Ahmed, N., Korkmaz, A., Rafiq, M., Baleanu, D., Alshomrani, A.S., Rehman, M.A., Iqbal, M.S.: A novel time efficient structure-preserving splitting method for the solution of two-dimensional reaction-diffusion systems. Adv. Diff. Eq. 2020(1), 1–26 (2020)
Odibat, Z.: Approximations of fractional integrals and Caputo fractional derivatives. Appl. Math. Comput. 178(2), 527–533 (2006)
Alqahtani, O., Karapinar, E., Shahi, P.: Common fixed point results in function weighted metric spaces. J. Inequal. Appl. 2019, 1–9 (2019)
Brooks, R. M., Schmitt, K.: THE CONTRACTION MAPPING PRINCIPLE AND SOME APPLICATIONS. Electron. J. Diff. Eq. 2009, (2009)
Kellogg, R.B.: Uniqueness in the Schauder fixed point theorem. Proc. Am. Math. Soc. 60(1), 207–210 (1976)
Jiang, X., Xu, M., Qi, H.: The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes. Nonlinear Anal.: Real World Appl. 11(1), 262–269 (2010)
Granas, A., Dugundji, J.: Fixed point theory (Vol. 14, pp. 15-16). Springer, New York (2003)
Torres, P.J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Diff. Eq. 190(2), 643–662 (2003)
Dowker, C.H.: Mapping theorems for non-compact spaces. Am. J. Math. 69(2), 200–242 (1947)
Adams, R. A., Fournier, J. J.: Sobolev spaces. Elsevier (2003)
Denardo, E.V.: Contraction mappings in the theory underlying dynamic programming. Siam Rev. 9(2), 165–177 (1967)
Jhangeer, A., Almusawa, H., Rahman, R.U.: Fractional derivative-based performance analysis to Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Results Phys. 36, 105356 (2022)
Iqbal, M. S., Baber, M. Z., Inc, M., Younis, M., Ahmed, N., Qasim, M.: On multiple solitons of glycolysis reaction-diffusion system for the chemical concentration. Int. J. Mod. Phys. B 2450055 (2023)
Baber, M. Z., Seadway, A. R., Ahmed, N., Iqbal, M. S., Yasin, M. W.: Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross-Pitaevskii equation in dispersive media. Int. J. Mod. Phys. B 2350191 (2022)
Younis, M., Seadawy, A.R., Baber, M.Z., Yasin, M.W., Rizvi, S.T., Iqbal, M.S.: Abundant solitary wave structures of the higher dimensional Sakovich dynamical model. Math. Methods Appl, Sci (2021)
Islam, W., Baber, M. Z., Ahmed, N., Akgül, A., Rafiq, M., Raza, A., ... Weera, W.: Investigation the soliton solutions of mussel and algae model leading to concentration. Alex. Eng. J. 70, 133–143 (2023)
Yasin, M.W., Iqbal, M.S., Seadawy, A.R., Baber, M.Z., Younis, M., Rizvi, S.T.: Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model. Int. J. Nonlinear Sci. Numer, Simul (2021)