Representation for the reproducing kernel Hilbert space method for a nonlinear system

Year 2019, Volume: 48 Issue: 5, 1345 - 1355, 08.10.2019

Esra Karatas Akgül Ali Akgül , Yasir Khan Dumitru Baleanu

Abstract

We apply the reproducing kernel Hilbert space method to a nonlinear system in this work. We utilize this  technique to overcome the nonlinearity of the problem. We obtain accurate results. We demonstrate our results by tables and figures. We prove the efficiency of the method.

Keywords

reproducing kernel functions, a nonlinear system, bounded linear operator

References

• [1] S. Abbasbandy, B. Azarnavid and M. S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, J. Comput. Appl. Math., 279, 293–305, 2015.
• [2] A. Akgul and M. Inc, Approximate solutions for mhd squeezing fluid flow by a novel method, Boundary Value Problems, 2014, Article number: 18, 2014.
• [3] B. Azarnavid and K. Parand, An iterative reproducing kernel method in Hilbert space for the multi-point boundary value problems, J. Comput. Appl. Math., 328, 151–163, 2018.
• [4] A. H. Bhrawy, M. A. Abdelkawy, E. M. Hilal, A. A. Alshaery and A. Biswas, Solitons, cnoidal waves, snoidal waves and other solutions to Whitham-Broer-Kaup system, Appl. Math. Inf. Sci., 8 (5), 2119–2128, 2014, doi:10.12785/amis/080505.
• [5] A. H. Bhrawy, J. F. Alzaidy, M. A. Abdelkawy and A. Biswas, Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations, Nonlinear Dynam., 84 (3), 1553–1567, 2016, doi:10.1007/s11071-015-2588-x.
• [6] A. Biswas, Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients, Nonlinear Dynam., 58 (1-2), 345–348, 2009, doi:10.1007/ s11071-009-9480-5.
• [7] D. Biswas and T. Banerjee, A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation, Nonlinear Dynam., 83 (4), 2331–2347, 2016, doi:10.1007/s11071-015-2484-4.
• [8] S. Choi and J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticle, ASME FED, 231, 99–105, 1995.
• [9] I. Cialenco, G. E. Fasshauer and Q. Ye, Approximation of stochastic partial differential equations by a kernel-based collocation method, Int. J. Comput. Math., 89 (18), 2543– 2561, 2012.
• [10] M. Cui and Y. Lin, Nonlinear numerical analysis in the reproducing kernel space, Nova Science Publishers Inc., New York, 2009.
• [11] G. E. Fasshauer, F. J. Hickernell and Q. Ye, Solving support vector machines in reproducing kernel Banach spaces with positive definite functions, Appl. Comput. Harmon. Anal., 38 (1), 115–139, 2015.
• [12] F. Geng and M. Cui, Solving a nonlinear system of second order boundary value problems, J. Math. Anal. Appl., 327 (2), 1167–1181, 2007, doi:10.1016/j.jmaa.2006. 05.011.
• [13] M. Inc, A. Akgul and A. Kilicman, Numerical solutions of the second-order onedimensional telegraph equation based on reproducing kernel Hilbert space method, Abstr. Appl. Anal., 2013, 13 pages, 2013.
• [14] R. Ketabchi, R. Mokhtari and E. Babolian, Some error estimates for solving volterra integral equations by using the reproducing kernel method, J. Comput. Appl. Math., 273, 245–250, 2015.
• [15] K. Khanafer, K. Vafai and M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Tran., 46, 3639– 3653, 2003.
• [16] M. Pantzali, A. Mouza and S. Paras, Investigating the efficacy of nanofluids as coolants in plate heat exchangers (phe), Chem. Eng. Sci., 64, 3290–3300, 2009.