Iterative Perturbation Technique for Solving a Special Magnetohydrodynamics Problem

Öz

Click here for manuscript sample template

In this paper, we use perturbation iteration technique for struggling magnetohydrodynamics Jeffery-Hamel flow problem. This problem aroused from the classical work by Navier and Stokes and their  equations. We exploit Maxwell’s electromagnetism governing equations via reducing them to nonlinear differential equations to reform the main problem. After simplifying the well-known equation, we get a basic problem and we can readily investigate the emerged problem.  In order to check the power of the technique, we prove that the results are well agreed with the numerical solutions. The present graphics prove that perturbation iteration technique has high accuracy for different α, Ha and Re numbers.

Anahtar Kelimeler

• Fluid mechanics,Jeffery-Hamel flows,perturbation iteration method

Kaynakça

1. 1. Eagles, P. 1966. The stability of a family of Jeffery-Hamel solutions for divergent channel flow. Journal of Fluid Mechanics; 24(1): 191-207.
2. 2. Jeffery, GB. 1915. The two-dimensional steady motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science; 29(172): 455-465.
3. 3. Hamel, G. 1917. Spiralförmige Bewegungen zäher Flüssigkeiten. Jahresbericht der deutschen mathematiker-vereinigung; 25: 34-60.
4. 4. Alfvén, H, Arrhenius, G. 1970. Structure and evolutionary history of the solar system. Astrophysics and Space Science; 8(3): 338-421.
5. 5. Esmaeilpour, M, Ganji, DD. 2010. Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method. Computers & Mathematics with Applications; 59(11): 3405-3411.
6. 6. He, J. H. 1999. Variational iteration method–a kind of non-linear analytical technique: some examples. International journal of non-linear mechanics; 34(4): 699-708.
7. 7. El-Tawil, MA, Bahnasawi, AA, Abdel-Naby, A. 2004. Solving Riccati differential equation using Adomian's decomposition method. Applied Mathematics and Computation; 157(2): 503-514.
8. 8. Liao, S. 2004. On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation; 147(2): 499-513.

9. 9. Marinca, V, Herişanu, N. 2008. Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. International Communications in Heat and Mass Transfer;35(6): 710-715.
10. 10. Pandir, Y. 2018. A New Type of Generalized F-Expansion Method and its Application to Sine-Gordon Equation. Celal Bayar Üniversitesi Fen Bilimleri Dergisi; 13(3): 647-650.
11. 11. Mollaoğlu, T, Sezer, M. 2017. A numerical approach with residual error estimation for eolution of high-order linear differential-difference equations by using Gegenbauer polynomials. Celal Bayar Üniversitesi Fen Bilimleri Dergisi; 13(1): 39-49.
12. 12. Şahin, M., Sezer, M. 2018. Pell-Lucas Collocation Method for Solving High-Order Functional Differential Equations with Hybrid Delays. Celal Bayar Üniversitesi Fen Bilimleri Dergisi; 14(2): 141-149.
13. 13. İnan, B. 2017. An Exponential Finite Difference Method Based on Padé Approximation. Celal Bayar Üniversitesi Fen Bilimleri Dergisi; 13(1): 71-80.
14. 14. Açil, M., Konuralp, A., Bildik, N. 2017. Finding The Lie Symmetries of Some First-Order Odes Via Induced Characteristic. Celal Bayar Üniversitesi Fen Bilimleri Dergisi; 13(2): 275-278.
15. 15. Bildik, N. 2017. General convergence analysis for the perturbation iteration technique. Turkish Journal of Mathematics and Computer Science; 6: 1-9.
16. 16. Bildik, N, Deniz, S. 2017. A new efficient method for solving delay differential equations and a comparison with other methods. The European Physical Journal Plus; 132(1), 51.
17. 17. Deniz, S. 2017. Optimal perturbation iteration method for solving nonlinear heat transfer equations. Journal of Heat Transfer; 139(7): 1-6.
18. 18. Deniz, S, Bildik, N. 2017. 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.
19. 19. Bildik, N, Deniz, S. 2018. New analytic approximate solutions to the generalized regularized long wave equations. Bulletin of the Korean Mathematical Society; 55(3): 749-762.
20. 20. Deniz, S, Bildik, N, Sezer, M. 2017. A note on stability analysis of Taylor collocation method. Celal Bayar University Journal of Science; 13(1): 149-153.
21. 21. Kahraman, T. (2018). Some Null Quaternionic Curves in Minkowski spaces. Celal Bayar University Journal of Science; 14 (4), 357-361.
22. 22. Kahraman, T. (2019). Null Quaternionic Slant Helices in Minkowski Spaces. Mathematical Combinatorics; 1, 45-52.
23. 23. Hoi, Y., Meng, H., Woodward, S. H., Bendok, B. R., Hanel, R. A., Guterman, L. R., & Hopkins, L. N. (2004). Effects of arterial geometry on aneurysm growth: three-imensional computational fluid dynamics study. Journal of neurosurgery; 101(4), 676-681.
24. 24. Akgül, A. (2014). Approximate solutions for MHD squeezing fluid flow by a novel method. Boundary Value Problems, 2014(1), 18.
25. 25. Akgül, A. (2019). Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a Powell–Eyring non-Newtonian fluid. Journal of Taibah University for Science, 13(1), 858-863.
26. 26. Hashemi, M. S., & Akgül, A. (2018). Solitary wave solutions of time–space nonlinear fractional Schrödinger’s equation: two analytical approaches. Journal of Computational and Applied Mathematics, 339, 147-160.
27. 27. Inc, M., Akgül, A., Kılıçman, A. (2013). A new application of the reproducing kernel Hilbert space method to solve MHD JefferyHamel flows problem in nonparallel walls. Abstract and Applied Analysis (Vol. 2013). Hindawi.
28. 28. Aksoy, Y., Pakdemirli, M. (2010). New perturbation–iteration solutions for Bratu-type equations. Computers & Mathematics with Applications, 59(8), 2802-2808.
29. 29. Aksoy, Y., Pakdemirli, M., Abbasbandy, S., Boyacı, H. (2012). New perturbation‐iteration solutions for nonlinear heat transfer equations. International journal of Numerical methods for Heat & fluid flow; 22(7), 814-828.
30. 30. Şenol, M., Timuçin Dolapçı, İ., Aksoy, Y., Pakdemirli, M. (2013). Perturbation-iteration method for first-order differential equations and systems. Abstract and Applied Analysis; (Vol. 2013). Hindawi.
31. 31. Deniz, S. (2020). Semi-analytical investigation of modified Boussinesq-Burger equations. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(1), 327-333.
32. 32. Bildik, N., Deniz, S. (2020). New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete & Continuous Dynamical Systems-S, 13(3), 503.
33. 33. Bildik, N., Deniz, S. (2020). A comparative study on solving fractional cubic isothermal auto-catalytic chemical system via new efficient technique. Chaos, Solitons & Fractals, 132, 109555.
34. 34. Agarwal, P., Deniz, S., Jain, S., Alderremy, A. A., Aly, S. (2020). A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Physica A: Statistical Mechanics and its Applications, 542, 122769.
35. 35. Bildik, N., Deniz, S. (2017). Modification of perturbation-iteration method to solve different types of nonlinear differential equations. AIP Conference Proceedings; 1798(1), 020027.
36. 36. Deniz, S., & Bildik, N. (2017). Applications of optimal perturbation iteration method for solving nonlinear differential equations. AIP Conference Proceedings; 1798(1), 020046.
37. 37. Bildik, N., Deniz, S. (2017). A practical method for analytical evaluation of approximate solutions of Fisher's equations. ITM Web of Conferences; 13, 01001.
38. 38. Bildik, N., Deniz, S. (2018). Solving the burgers' and regularized long wave equations using the new perturbation iteration technique. Numerical Methods for Partial Differential Equations, 34(5), 1489-1501.