On the Semi-Analytical Solutions for the Kudryashov-Sinelshchikov Dynamical Equation Arising in Mixtures of Liquid and Gas Bubbles Without Neglecting of Heat Transfer and Viscosity

Abstract

In this study, Banach contraction method (BCM), Daftardar-Jafari method (DJM) and modified variational iteration method (MVIM) are proposed for the semi-analytical solutions of the Kudryashov-Sinelshchikov (K-S) dynamical equation. It has been shown that the analytical and semi-analytical solutions for the K-S dynamical equation with initial value problems by using semi-analytical methods can be obtained. In addition, the effectiveness and usefulness of the semi-analytical methods used are supported by tables and 3D figures. As the number of iteration or terms increases, how the semi-analytical solutions behave over time and converge to the exact solution is shown in detail with 2D figures. Also, it is shown comparison of semi-analytical solutions with exact solutions and error analysis with the help of tables. It has been discussed the methods are compared with each other and whether they are suitable for the K-S dynamical equation.

Keywords

• Kudryashov-Sinelshchikov (K-S) dynamical equation
• Banach contraction method (BCM)
• Daftardar- Jafari method (DJM)
• modified variational iteration method (MVIM)

References

1. Abassy, T.A., El Tawil, M., El Zoheiry, H., Toward a modified variational iteration method , Journal of Computational and Applied Mathematics, 207(2007), 137–147.
2. Abassy, T.A., Modified variational iteration method (nonlinear homogeneous initial value problem), Computers and Mathematics with Applications, 59(2010), 912–918.
3. Abbasbandy, S., Shivanian, E., Application of the variational iteration method for system of nonlinear Volterra’s integro-differential equations, Mathematical and Computational Applications, 14(2009), 147–158.
4. Abed, M.S., Al-Jawary, M.A.,Efficient iterative methods for solving the SIR epidemic model, Iraqi Journal of Science, 62(2021), 613–622.
5. Al-Jawary, M.A., Analytical solutions for solving fourth-order parabolic partial differential equations with variable coefficients, International Journal of Advances Scientific and Technical Research,3(2015), 531–545.
6. Al-Jawary, M.A., Abd-Al-Razaq, S.G., Analytical and numerical solution for duffing equations, International Journal of Basic and Applied Sciences, 5(2016), 115–119.
7. Al-Jawary, M.A., Adwan, M.I., Reliable iterative methods for solving the Falkner-Skan equation , Gazi University Journal of Science, 33(2020), 168–186.
8. Al- Jawary, M.A., Nabi, Al-Z. J.A., Three iterative methods for solving Jeffery-Homel flow problem, Kuwait Journal of Science, 47(2020), 1–13.

9. Almjeed, S.H., The approximate solution of the Fornberg-Whitham equation by a semi-analytical iterative technique, Engineering and Technology Journal, 36(2018), 120–123.
10. Başkonuş, H.M., Mahmud, A.A., Abdulrahman, K., Tanrıverdi, T., Gao,W., Studying on Kudryashov-Sinelshchikov dynamical equation arising in mixtures of liquid and gas bubbles, Thermal Science, 26(2022), 1229–1244.
11. Bhalekar, S., Patade, J., Analytical solutions of nonlinear equations with proportional delays, Applied and Computational Mathematics, 15(2016), 331–345.
12. Chen, C., Rui, W., Long, Y., Different kinds of singular and nonsingular exact traveling wave solutions of the Kudryashov-Sinelshchikov equation in the special parametric conditions, Mathematical Problems in Engineering, article id: 456964(2013), 10 pages.
13. Dafdardar-Gejji, V., Bhalekar, S., Solving nonlinear functional equation using Banach contraction principle, Far East Journal of Applied Mathematics, 34(2009), 303–314.
14. Dafdardar-Gejji, V., Jafari, H., An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, 316(2006), 753–763.
15. Easif, F.H., Manaa, S.A., Sabali, A.J., Modified variational iteration method and homotopy analysis method for solving variable coefficient variant Boussinesq system, General Letters in Mathematics, 8(2020), 26–32.
16. Ghitheeth, A.E., Mahmood, H.S., Solve partial differential equations using the Banach contraction method and improve results using the trapezoidal rule, Al-Rafidain Journal of Computer Sciences and Mathematics, 15(2021), 79–85.
17. Güner, O., Bekir, A., Çevikel, A.C., Dark soliton and periodic wave solutions of nonlinear evolution equations, Advances in Difference Equations, 2013(2013),11 pages.
18. He, Y.,New Jacobi elliptic function solutions for the Kudryashov-Sinelshchikov equation using improved F-expansion method, Mathematical Problems in Engineering, article id:104894(2013), 6 pages.
19. He, Y., Li, S., Long, Y., Exact solutions of the Kudryashov-Sinelshchikov equation using the multiple (G’/G)-expansion method, Mathematical Problems in Engineering, article id:708049(2013), 7 pages.
20. Inc¸, M., Khan, H., Baleanu, D., Khan, A., Modified variational iteration method for straight fins with temperature dependent thermal conductivity, Thermal Science, 22(2018), 229–236.
21. Kaplan, M., Bekir, A., Akbulut, A.,Analytical solutions with the improved (G’/G)-expansion method for nonlinear evolution equations, Journal of Physics: Conference Series, , 766(2016), 012033.
22. Köprülü, M.O., Investigation off exact solutions of some nonlinear evolution equation via an analytical approach, Mathematical Science and Applications E-Notes, 9(2021), 64–73.
23. Kudryashov, N.A., Sinelshchikov, D.I., Nonlinear wave in bubby liquids with consideration for viscosity and heat transfer, Physics Letters A, 374(2010˙I90)(2010), 2011–2016.
24. Kumar, A., Methi, G., An efficient numerical algorithm for solution of nonlinear delay differential equations, Journal of Physics: Conference Series,1849(2021), 012014, 9 pages.
25. Lu, J., New exact solutions for Kudryashov-Sinelshchikov equation, Advance in Difference Equations, 2018(2018), 1–17.
26. Lu, J-F., Modified variational iteration method for variant Boussinesq equation , Thermal Science, 19(2015), 1195–1199.
27. Nabi, Al-Z.A., Al-Jawary, M., Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems , Gazi University Journal of Science, 32(2019), 967–989.
28. Ogundile, O.P., Edeki, S.O., Olaniregun, D., G., Iterative methods for solving Riccati differential equations, Journal of Physics: Conference Series, 1734(2021), 012003, 6 pages.
29. Ryabov, P.N., Exact solutions of the Kudryashov-Sinelshchikov equation, Applied Mathematics and Computation, 217(2010), 3585–3590.
30. Seadawy, A.R., Iqbal, M., Lu, D., Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity, Journal of Taibah University for Science, 13(2019), 1060–1072.
31. Subhaschandra, S., Solutions of Kudryashov-Sinelshchikov equation and generalized Radhakrishnan-Kundu-Lakshmanan equation by the first integral method, International Journal of Physical Research,4(2016), 37–42.
32. Yusuh, A., Inc¸, M., Bayram, M., Soliton solutions for Kudryashov-Sinelshchikov equation, Sigma Journal of Engineering and Natural Sciences, 37(2019), 439–444.
33. Zhao, Y-M., F-expansion method and its application for finding exact solutions to the Kudryashov-Sinelshchikov equation, Journal of Applied Mathematics, article id:895760(2013), 7 pages.