Semi-analytical investigation of modified Boussinesq-Burger equations

Abstract

In this paper, a new analysis of nonlinear modified Boussinesq-Burger equation is revisited via optimal perturbation iteration technique. We first consider artificial parameters and perturbation theory and combine them to deal with nonlinear partial differential equations. After that, the recommended theory is employed to get new semi-analytical solutions of nonlinear partial differential equations. As will be seen from the results, this technique needs no discretization or linearization and can be directly applied to many nonlinear differential equations.

Keywords

• Optimal perturbation iteration techniques,nonlinear partial differential equations

Değiştirilmiş Boussinesq-Burger denklemlerinin yarı analitik incelemesi

Abstract

Bu çalışmada değiştirilmiş Boussinesq-Burger denklemlerinin optimal perturbasyon iterasyon yöntemi ile yarı analitik incelemesi yapılmıştır. Öncelikle önerilen metodun inşası için yapay parametreler ve perturbasyon teorisi ele alınmış ve bunlar birleştirilerek lineer olmayan kısmi diferansiyel denklemler için bir çözüm metodu geliştirilmiştir. Daha sonra ise elde edilen algoritmalar ile ele alınan problem yarı analitik olarak çözülmüştür. Sonuçlardan da anlaşılabileceği üzere bu teknik birçok lineer olmayan diferansiyel denkleme herhangi bir lineerizasyon gerektirmeden rahatlıkla uygulanabilmektedir.

Keywords

• Optimal perturbasyon iterasyon yöntemi,lineer olmayan kısmi diferansiyel denklemler

References

1. Wadati, M., Heiji, S. and Kimiaki K., Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Progress of Theoretical Physics, 53(2), 419-436, (1975).
2. Bildik, N. and Deniz, S., Modified Adomian decomposition method for solving Riccati differential equations, Review of the Air Force Academy, 3(30), 21-26, (2015).
3. Boling, G., Liming, L. and Liu, Q., Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions, Physical Review E, 85(2), 026607, (2012).
4. Mohamed, M. and Saad, K., A numerical study by using the Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation, International Journal of Biomathematics, 11(8), 1850099, (2018).
5. Wazwaz, A.M., Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Applied Mathematics and Computation, 190(1), 633-640, (2007).
6. Bildik, N. and Deniz, S., Applications of Taylor collocation method and Lambert W function to the systems of delay differential equations, Turkish Journal of Mathematics and Computer Sciences 1(1), 1-13, (2013).
7. Engui, F. and Zhang, H., A note on the homogeneous balance method, Physics Letters A, 246(5), 403-406, (1998).
8. Bildik, N. and Deniz, S., Comparison of solutions of systems of delay differential equations using Taylor collocation method, Lambert W function and variational iteration method, Scientia Iranica. Transaction D, Computer Science & Engineering and Electrical Engineering, 22(3),1052-1058, (2015).

9. Sarp, Ü., Evirgen, F. and İkikardeş, S., Applications of differential transformation method to solve systems of ordinary and partial differential equations. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(2), 135-156, (2018).
10. Evirgen, F. and Özdemir, N., A fractional order dynamical trajectory approach for optimization problem with HPM, Fractional Dynamics and Control, Springer, New York, 145-155, (2012).
11. Deniz, S. and Bildik, N. Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University-Science, 30(1), 91-99, (2018).
12. Bildik, N. and Deniz, S., A new efficient method for solving delay differential equations and a comparison with other methods, The European Physical Journal Plus, 132(1), 51, (2017).
13. Deniz, S., Optimal perturbation iteration method for solving nonlinear heat transfer equations, Journal of Heat Transfer-ASME, 139(37), 074503-1, (2017).
14. Deniz, S. and Bildik, N., A new analytical technique for solving Lane - Emden type equations arising in astrophysics, Bulletin of the Belgian Mathematical Society – Simon Stevin, 24(2), 305-320, (2017).
15. Bildik, N. and Deniz, S., Solving the Burgers’ and regularized long wave equations using the new perturbation iteration technique, Numerical Methods for Partial Differential Equations, 34(5), 1489-1501, (2018).
16. Bildik, N. and Deniz, S., New analytic approximate solutions to the generalized regularized long wave equations, Bulletin of the Korean Mathematical Society, 55(3), 749-762, (2018).
17. Bildik, N. and Deniz, S., A new fractional analysis on the polluted lakes system, Chaos, Solitons & Fractals, 122, 17-24, (2019).
18. Eskitaşçıoğlu, E., Aktaş, M. and Baskonus H.M., New complex and hyperbolic forms for Ablowitz–Kaup–Newell–Segur wave equation with fourth order, Applied Mathematics and Nonlinear Sciences, 4(1), 105-112, (2019).
19. Yel, G., Baskonus, H.M., and Bulut, H., Regarding some novel exponential travelling wave solutions to the Wu–Zhang system arising in nonlinear water wave model, Indian Journal of Physics, 93,8, 1031-1039, (2019).
20. Baskonus, H.M., Complex soliton solutions to the Gilson–Pickering model, Axioms, 8(1), 18, (2019).
21. Ebadi, G., Envelope solitons, periodic waves and other solutions to Boussinesq-Burgers equation, Romanian Reports in Physics, 64(4), 915-932, (2012).