×

A new application of the reproducing kernel Hilbert space method to solve MHD Jeffery-Hamel flows problem in nonparallel walls. (English) Zbl 1383.76547

Summary: The present paper emphasizes Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is \(2\alpha\). A new method called the reproducing kernel Hilbert space method (\(RKHSM\)) is briefly introduced. The validity of the reproducing kernel method is set by comparing our results with HAM, DTM, and HPM and numerical results for different values of \(H\), \(\alpha\), and Re. The results show up that the proposed reproducing kernel method can achieve good results in predicting the solutions of such problems. Comparison between obtained results showed that \(RKHSM\) is more acceptable and accurate than other methods. This method is very useful and applicable for solving nonlinear problems.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Jeffery, G. B., The two-dimensional steady motion of a viscous fluid, Philosophical Magazine, 29, 172, 455-465, (1915) · JFM 45.1088.01
[2] Hamel, G.; Bewgungen, S.; Flussigkeiten, Z., Jahresbericht der Deutschen, Mathematiker-Vereinigung, 25, 34-60, (1916)
[3] Moghimi, S. M.; Domairry, G.; Soleimani, S.; Ghasemi, E.; Bararnia, H., Application of homotopy analysis method to solve MHD Jeffery-Hamel flows in non-parallel walls, Advances in Engineering Software, 42, 3, 108-113, (2011) · Zbl 1316.76078
[4] Esmaeilpour, M.; Ganji, D. D., Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method, Computers & Mathematics with Applications, 59, 11, 3405-3411, (2010) · Zbl 1197.76043
[5] Joneidi, A. A.; Domairry, G.; Babaelahi, M., Three analytical methods applied to Jeffery-Hamel flow, Communications in Nonlinear Science and Numerical Simulation, 15, 11, 3423-3434, (2010)
[6] Moghimi, S. M.; Ganji, D. D.; Bararnia, H.; Hosseini, M.; Jalaal, M., Homotopy perturbation method for nonlinear MHD Jeffery-Hamel problem, Computers & Mathematics with Applications, 61, 8, 2213-2216, (2011) · Zbl 1219.76038
[7] Esmaili, Q.; Ramiar, A.; Alizadeh, E.; Ganji, D. D., An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method, Physics Letters, 372, 19, 3434-3439, (2008) · Zbl 1220.76035
[8] Motsa, S. S.; Sibanda, P.; Awad, F. G.; Shateyi, S., A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem, Computers & Fluids, 39, 7, 1219-1225, (2010) · Zbl 1242.76363
[9] Goldstein, S., Modem Developments in Fluid Dynamics, 1, (1938), Oxford, UK: Clarendon Press, Oxford, UK
[10] Axford, W. I., The magnetohydrodynamic Jeffrey-Hamel problem for a weakly conducting fluid, The Quarterly Journal of Mechanics and Applied Mathematics, 14, 335-351, (1961) · Zbl 0106.40801
[11] Abbasbandy, S.; Shivanian, E., Exact analytical solution of the MHD Jeffery-Hamel fow problem, Mecannica, 47, 6, 1379-1389, (2012) · Zbl 1293.76162
[12] Makinde, O. D., Effect of arbitrary magnetic Reynolds number on MHD flows in convergent-divergent channels, International Journal of Numerical Methods for Heat & Fluid Flow, 18, 5-6, 697-707, (2008) · Zbl 1231.76353
[13] Makinde, O. D.; Mhone, P. Y., Hermite-Padé approximation approach to MHD Jeffery-Hamel flows, Applied Mathematics and Computation, 181, 2, 966-972, (2006) · Zbl 1102.76049
[14] Aronszajn, N., Theory of reproducing kernels, Transactions of the American Mathematical Society, 68, 337-404, (1950) · Zbl 0037.20701
[15] Cui, M.; Lin, Y., Nonlinear Numerical Analysis in the Reproducing Kernel Space, (2009), New York, NY, USA: Nova Science Publishers, New York, NY, USA · Zbl 1165.65300
[16] Geng, F.; Cui, M., Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications, 327, 2, 1167-1181, (2007) · Zbl 1113.34009
[17] Geng, F., A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Applied Mathematics and Computation, 213, 1, 163-169, (2009) · Zbl 1166.65358
[18] Geng, F.; Cui, M., New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions, Journal of Computational and Applied Mathematics, 233, 2, 165-172, (2009) · Zbl 1205.65216
[19] Geng, F.; Cui, M.; Zhang, B., Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods, Nonlinear Analysis, 11, 2, 637-644, (2010) · Zbl 1187.34012
[20] Geng, F.; Cui, M., Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems, Journal of Computational and Applied Mathematics, 235, 8, 2405-2411, (2011) · Zbl 1209.65078
[21] Geng, F.; Cui, M., A novel method for nonlinear two-point boundary value problems: combination of ADM and RKM, Applied Mathematics and Computation, 217, 9, 4676-4681, (2011) · Zbl 1208.65103
[22] Mohammadi, M.; Mokhtari, R., Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, Journal of Computational and Applied Mathematics, 235, 14, 4003-4014, (2011) · Zbl 1220.65143
[23] Jiang, W.; Lin, Y., Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Communications in Nonlinear Science and Numerical Simulation, 16, 9, 3639-3645, (2011) · Zbl 1223.35112
[24] Wang, Y.; Su, L.; Cao, X.; Li, X., Using reproducing kernel for solving a class of singularly perturbed problems, Computers & Mathematics with Applications, 61, 2, 421-430, (2011) · Zbl 1211.65142
[25] Wu, B. Y.; Li, X. Y., A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method, Applied Mathematics Letters, 24, 2, 156-159, (2011) · Zbl 1215.34014
[26] Yao, H.; Lin, Y., New algorithm for solving a nonlinear hyperbolic telegraph equation with an integral condition, International Journal for Numerical Methods in Biomedical Engineering, 27, 10, 1558-1568, (2011) · Zbl 1232.65135
[27] Geng, F.; Cui, M., A reproducing kernel method for solving nonlocal fractional boundary value problems, Applied Mathematics Letters, 25, 5, 818-823, (2012) · Zbl 1242.65144
[28] Inc, M.; Akgül, A., The reproducing kernel hilbert space method for solving troesch’s problem, Journal of the Association of Arab Universities For Basic and Applied Sciences, (2013)
[29] Inc, M.; Akgül, A.; Geng, F., Reproducing kernel hilbert space method for solving bratu’s problem · Zbl 1320.34017
[30] Inc, M.; Akgül, A.; Kiliçman, A., Explicit solution of telegraph equation based on reproducing kernel method, Journal of Function Spaces and Applications, 2012, (2012) · Zbl 1259.35062
[31] Inc, M.; Akgül, A.; Kiliçman, A., A novel method for solving KdV equation based on reproducing Kernel Hilbert space method, Abstract and Applied Analysis, 2013, (2013) · Zbl 1266.65178
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.