A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators

Authors

DOI:

https://doi.org/10.11121/ijocta.2021.1177

Keywords:

Force KdV equation, Fractional derivatives, q-Homotopy analysis transform technique, Fixed point theorem

Abstract

The Korteweg–De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface critical flow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional operators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physical nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order.

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Author Biographies

Pundikala Veeresha, Center for Mathematical Needs, Department of Mathematics, CHRIST (Deemed to be University), Bengaluru-560029, India

is an Assistant Professor in the Department of Mathematics, CHRIST (Deemed to be University), Bengaluru. He completed his Master's Degree from Davangere University, Davangere, and his Ph.D. from Karnatak University, Dharwad. His areas of research interest includes Fractional Calculus, Mathematical Modelling, Numerical and Analytical Methods, and Mathematical Physics. He has been the author of more than 70 research articles published in highly reputed journals.

Mehmet Yavuz, Necmettin Erbakan University, Faculty of Science, Department of Mathematics and Computer Sciences, 42090, Konya, Turkey

received his Ph.D. in Mathematics from Balıkesir University, Turkey. He visited the University of Exeter, U.K. for post-doctoral research in mathematical biology and optimal control theory for a year. He is currently serving as an associate professor at Necmettin Erbakan University, Turkey. His research interests mainly focus on infectious disease dynamics, fractional mathematical modeling, fractional theory and method, optimal control theory and bifurcation analysis. He has published more than 60 research papers in international esteemed journals and he is a reviewer for about seventy international repute journals as well as he is Editor-in-Chief of the "Mathematical Modelling and Numerical Simulation with Applications" journal.

Chandrali Baishya, Department of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, India

has obtained her PhD degree in Mathematical Sciences from the University of Mysore, Karnataka, India in 2009. She has been working as Assistant Professor in Tumkur University, Karnataka, India since 2011. Her areas of research interest are numerical analysis, fractional differential equations and mathematical biology.

References

Liouville, J. (1832). Memoire surquelques questions de geometrieet de mecanique, et sur un nouveau genre de calcul pour resoudreces questions, J. Ecole. Polytech., 13, 1-69.

Riemann, G.F.B. (1896). Versuch Einer Allgemeinen Auffassung der Integration und Differentiation, Gesammelte Mathematische Werke, Leipzig.

Caputo, M. (1969). Elasticita e Dissipazione, Zanichelli, Bologna.

Miller, K.S. & Ross, B. (1993). An introduction to fractional calculus and fractional differential equations, A Wiley, New York.

Podlubny, I. (1999) Fractional Differential Equations, Academic Press, New York.

Kilbas, A.A., Srivastava, H.M. & Trujillo, J.J. (2006). Theory and applications of fractional differential equations, Elsevier, Amsterdam.

Baleanu, D. Guvenc, Z.B. & Tenreiro Machado, J.A. (2010). New trends in nanotechnology and fractional calculus applications, Springer Dordrecht Heidelberg, London New York .

Naik, P. A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. (2020). Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42.

Evirgen, F., & Yavuz, M. (2018). An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative. In ITM Web of Conferences (Vol. 22, p. 01009). EDP Sciences.

Yokus, A. (2021). Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrodinger equation. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 24-31.

Yavuz, M., Cosar, F.O., Gunay, F., & Ozdemir, F. N. (2021). A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign. Open Journal of Modelling and Simulation, 9(3), 299-321.

Ucar, E., Ucar, S., Evirgen, F., & Ozdemir, N. (2021). A Fractional SAIDR Model in the Frame of Atangana-Baleanu Derivative. Fractal and Fractional, 5(2), 32.

Kumar, P., Erturk, V. S., Banerjee, R., Yavuz, M., & Govindaraj, V. (2021). Fractional modeling of plankton-oxygen dynamics under climate change by the application of a recent numerical algorithm. Physica Scripta, 96(12), 124044.

Dasbas, B. (2021). Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 44-55.

Ucar, E., Ucar, S., Evirgen, F., & Ozdemir, N. (2021). Investigation of E-Cigarette Smoking Model with Mittag-Leffler Kernel. Foundations of Computing and Decision Sciences, 46(1), 97-109.

Akinyemi, L. et al., (2021), Novel soliton solutions of four sets of generalized (2+1)-dimensional Boussinesq- Kadomtsev-Petviashvili-like equations, Modern Physics Letters B, 2150530, DOI: 10.1142/s0217984921505308.

Veeresha, P. (2021). A numerical approach to the coupled atmospheric ocean model using a fractional operator. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 1-10.

Baishya, C. & Veeresha, P. (2021), Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel, Proceeding of the Royal Society A, 477(2253).

Baishya, C., Achar, S.J., Veeresha, P. & Prakasha, D.G. (2021), Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos, 31, 043130.

Baskonus, H.M., T. A. Sulaiman, H. Bulut, On the new wave behavior to the Klein-Gordon-Zakharov equations in plasma physics, Indian J. Phys. 93 (3) (2019) 393- 399.

Yao, S.W., Ilhan, E., Veeresha, P. & Baskonus, H.M. (2021). A powerful iterative approach for quintic complex Ginzburg-Landau equation within the frame of fractional operator, Fractals, DOI: 10.1142/S0218348X21400235.

Veeresha, P., Prakasha, D.G. & Baleanu, D. (2019). An effcient numerical technique for the nonlinear fractional Kolmogorov- Petrovskii-Piskunov equation, Mathematics, 7(3), 1-18. DOI:10.3390/math7030265.

Ali, M.R., Ma, W.X. & Sadat, R. (2021). Lie symmetry analysis and invariant solutions for (2+1) dimensional Bogoyavlensky-Konopelchenko equation with variable-coeffcient in wave propagation. Journal of Ocean Engineering and Science, DOI: 10.1016/j.joes.2021.08.006.

Akinyemi, L., & Iyiola, O. S. (2021). Analytical Study of (3 + 1)-Dimensional Fractional- Reaction Diffusion Trimolecular Models. International Journal of Applied and Computational Mathematics, 7(3), 1-24.

Veeresha, P. & Baleanu, D. (2021). A unifying computational framework for fractional Gross-Pitaevskii equations. Physica Scripta, 96(125010).

Ali, M.R., Sadat, R. & Ma, W.X. (2021). Investigation of new solutions for an extended (2+1)-dimensional Calogero-Bogoyavlenskii- Schif equation. Frontiers of Mathematics in China, 16(4), 925-936.

Akinyemi, L., & Iyiola, O.S. (2020). A reliable technique to study nonlinear time fractional coupled Korteweg-de Vries equations. Advances in Difference equations, 2020(1), 1-27.

Ali, M.R. & Ma, W.X. (2020). New exact solutions of Bratu Gelfand model in two dimensions using Lie symmetry analysis. Chinese Journal of Physics, 65, 198-206.

Akinyemi, L., et al. (2021). Novel approach to the analysis of fifth-order weakly nonlocal fractional Schrodinger equation with Caputo derivative. Results in Physics, 104958, DOI: 10.1016/j.rinp.2021.104958.

Safare, K.M. et al. (2021). A mathematical analysis of ongoing outbreak COVID-19 in India through nonsingular derivative. Numer- ical Methods for Partial Differential Equations, 37(2), 1282-1298.

Akinyemi, L., Veeresha, P., & Ajibola, S.O. (2021). Numerical simulation for coupled nonlinear Schrodinger-Korteweg-de Vries and Maccari systems of equations. Modern Physics Letters B, 2150339.

Hammouch, Z., Yavuz, M., & Ozdemir, N. (2021). Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 11-23.

Veeresha, P., Ilhan, E., Prakasha, D.G., Baskonus, H.M. & Gao, W. (2021). Regarding on the fractional mathematical model of Tumour invasion and metastasis. Computer Modeling in Engineering & Sciences, 127(3), 1013-1036.

Akinyemi, L., & Huseen, S.N. (2020). A powerful approach to study the new modiffed coupled Korteweg-de Vries system. Mathematics and Computers in Simulation, 177, 556-567.

Ali, M.R., & Sadat, R. (2021). Construction of Lump and optical solitons solutions for (3+ 1) model for the propagation of nonlinear dispersive waves in inhomogeneous media. Optical and Quantum Electronics, 53(5), 1-13.

Baishya, C. (2021). An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative. SeMA Journal, 1-19.

Senol, M., Iyiola, O.S., Daei Kasmaei, H., & Akinyemi, L. (2019). Effcient analytical techniques for solving time-fractional nonlinear coupled Jaulent-Miodek system with energydependent Schrodinger potential. Advances in Difference Equations, 2019(1), 1-21.

Veeresha, P., Ilhan, E., Prakasha, D.G., Baskonus, H.M. & Gao, W. (2021). A new numerical investigation of fractional order susceptible-infected-recovered epidemic model of childhood disease. Alexandria Engineering Journal, DOI: 10.1016/j.aej.2021.07.015.

Ali, M.R., & Sadat, R. (2021). Lie symmetry analysis, new group invariant for the (3+ 1)- dimensional and variable coeffcients for liquids with gas bubbles models. Chinese Journal of Physics, 71, 539-547.

Akinyemi, L., Senol, M., & Huseen, S. N. (2021). Modified homotopy methods for generalized fractional perturbed Zakharov-Kuznetsov equation in dusty plasma. Advances in Difference Equations, 2021(1), 1- 27.

Veeresha, P. & Prakasha, D.G. (2021). Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional operator. Communications in Theoretical Physics, 72(10).

Dias, F. & Vanden-Broeck, J.M. (2002). Generalized critical free-surface ows, J. Eng. Math. 42, 291-301.

Shen, S.S. (1995). On the accuracy of the stationary forced Korteweg-De Vries equation as a model equation for ows over a bump, Q. Appl. Math. 53, 701-719.

Camassa, R. & Wu, T. (1991), Stability of forced solitary waves, Philos. Trans. R. Soc. Lond. A, 337, 429-466.

Zabuski, N.J. & Kruskal, M.D. (1965). Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240-243.

Crighton, D.G. (1995). Applications of KdV, Acta Appl. Math., 39, 39-67.

Hereman, W. (2012), Shallow Water Waves and Solitary Waves, In Mathematics of Complexity and Dynamical Systems; Meyers, R., Ed.; Springer: New York, USA.

Yao-Tsu Wu, T. (1987). Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech., 184, 75-99.

David, V.D., Aziz, Z.A. & Salah, F. (2016). Analytical approximate solution for the forced Korteweg-de Vries (FKdV) on critical ow over a hole using homotopy analysis method, Jurnal Teknologi (Sciences & Engineering), 78(3-2), 107-112.

Caputo, M. & Fabrizio, M. (2016). A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), 73-85.

Atangana, A. & Baleanu, D. (2016). New fractional derivatives with non-local and nonsingular kernel theory and application to heat transfer model, Thermal Science, 20, 763-769.

Jun-Xiao, Z. & Bo-Ling, G. (2009), Analytic solutions to forced KdV equation. Commun. Theor. Phys., 52, 279-283.

Milewski, P.A. (2014). The forced Korteweg- De Vries equation as a model for waves generated by topography. Cubo, 6(4), 33-51.

David, V.D., Salah, F., Nazari, M. & Aziz, Z.A. (2013). Approximate analytical solution for the forced Korteweg-de Vries equation. Journal of Applied Mathematics, 1-9, DOI: 10.1155/2013/795818.

Lee, S. (2018). Dynamics of trapped solitary waves for the forced KdV equation, Symmetry 10(129), 1-13, DOI: 10.3390/sym10050129.

Tay, K.G., Tiong, W.K., Choy, Y.Y. & Ong, C.T. (2017). Method of lines and pseudospectral solutions of the forced Korteweg-De Vries equation with variable coefficients arises in elastic tube. International Journal of Pure and Applied Mathematics, 116(4), 985-999.

Liao, S.J. (1997). Homotopy analysis method and its applications in mathematics, J. Basic Sci. Eng., 5(2), 111-125.

Liao, S.J. (1998). Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech., 19, 957-962.

Singh, J., Kumar, D. & Swroop, R. (2016). Numerical solution of time and space fractional coupled Burgers' equations via homotopy algorithm. Alexandria Eng. J., 55(2), 1753-1763.

Srivastava, H.M., Kumar, D. & Singh, J., An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model., 45, 192-204.

Veeresha, P., Prakasha, D.G., Singh, J., Kumar, D. & Baleanu, D. (2020). Fractional Klein-Gordon-Schrodinger equations with Mittag-Leffler memory. Chinese J. Phy., 68, 65-78.

Prakasha, D.G., et al. (2021). An ef- cient computational technique for time fractional Kaup-Kupershmidt equation. Numerical Methods for Partial Differential Equations, 37(2), 1299-1316.

Veeresha, P. & Prakasha, D.G. (2019). A novel technique for (2+1)-dimensional time fractional coupled Burgers equations. Mathematics and Computers in Simulation, 166, 324-345.

D. Kumar, R.P. Agarwal, J. Singh, A modified numerical scheme and convergence analysis for fractional model of Lienard's equation, J. Comput. Appl. Math. 399 (2018) 405-413.

Veeresha, P., Ilhan, E. & Baskonus, H.M. (2021). Fractional approach for analysis of the model describing wind-in uenced projectile motion. Physica Scripta, 96(7), 075209.

Veeresha, P. & Prakasha, D.G. (2021). Solution for fractional Kuramoto-Sivashinsky equation using novel computational technique. International Journal of Applied and Computational Mathematics, 7(33).

Okposo, N.I., Veeresha, P. & Okposo, E.N. (2021). Solutions for time-fractional coupled nonlinear Schrodinger equations arising in optical solitons. Chinese Journal of Physics, DOI: 10.1016/j.cjph.2021.10.014.

Losada, J. & Nieto, J.J. (2015), Properties of the new fractional derivative without singular Kernel, Progr. Fract. Differ. Appl., 1, 87-92.

Atangana, A. & Koca, I. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals, 89, 447-454.

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Published

2021-12-31
CITATION
DOI: 10.11121/ijocta.2021.1177
Published: 2021-12-31

How to Cite

Veeresha, P., Yavuz, M., & Baishya, C. . (2021). A computational approach for shallow water forced Korteweg–De Vries equation on critical flow over a hole with three fractional operators. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 11(3), 52–67. https://doi.org/10.11121/ijocta.2021.1177

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Research Articles