×

Solutions of a class of sixth order boundary value problems using the reproducing kernel space. (English) Zbl 1364.65148

Summary: The approximate solution to a class of sixth order boundary value problems is obtained using the reproducing kernel space method. The numerical procedure is applied on linear and nonlinear boundary value problems. The approach provides the solution in terms of a convergent series with easily computable components. The present method is simple from the computational point of view, resulting in speed and accuracy significant improvements in scientific and engineering applications.It was observed that the errors in absolute values are better than compared (C. H. C. Hussin and A. Kiliçman [Math. Probl. Eng. 2011, Article ID 724927, 19 p. (2011; Zbl 1213.65148)] and M. A. Noor and S. T. Mahyud-Din [Comput. Math. Appl. 55, No. 12, 2953–2972 (2008; Zbl 1142.65386)], A.-M. Wazwaz [Appl. Math. Comput. 118, 311–325 (2001; Zbl 1023.65074)], P. K. Pandey [Int. J. Pure Appl. Math. 76, No. 3, 317–326 (2012; Zbl 1250.65102)]. Furthermore, the nonlinear boundary value problem for the integro-differential equation has been investigated arising in chemical engineering, underground water flow and population dynamics, and other fields of physics and mathematical chemistry. The performance of reproducing kernel functions is shown to be very encouraging by experimental results.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34K10 Boundary value problems for functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Toomre, J.; Zahn, J. R.; Latour, J.; Spiegel, E. A., Stellar convection theory ii:single mode study of the second convection zone in a-type stars, The Astrophysical Journal, 207, 545-563 (1976) · doi:10.1086/154522
[2] Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability. Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics (1961), Oxford, UK: Clarendon Press, Oxford, UK · Zbl 0142.44103
[3] Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations, xii+307 (1986), Singapore: World Scientific Publishing, Singapore · Zbl 0619.34019
[4] Akram, G.; Siddiqi, S. S., Solution of sixth order boundary value problems using non-polynomial spline technique, Applied Mathematics and Computation, 181, 1, 708-720 (2006) · Zbl 1155.65361 · doi:10.1016/j.amc.2006.01.053
[5] Siddiqi, S. S.; Akram, G., Septic spline solutions of sixth-order boundary value problems, Journal of Computational and Applied Mathematics, 215, 1, 288-301 (2008) · Zbl 1138.65062 · doi:10.1016/j.cam.2007.04.013
[6] Noor, M. A.; Mohyud-Din, S. T., Homotopy perturbation method for solving sixth-order boundary value problems, Computers & Mathematics with Applications, 55, 12, 2953-2972 (2008) · Zbl 1142.65386 · doi:10.1016/j.camwa.2007.11.026
[7] Wazwaz, A.-M., The numerical solution of sixth-order boundary value problems by the modified decomposition method, Applied Mathematics and Computation, 118, 2-3, 311-325 (2001) · Zbl 1023.65074 · doi:10.1016/S0096-3003(99)00224-6
[8] Che Hussin, C. H.; Kiliçman, A., On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method, Mathematical Problems in Engineering, 2011 (2011) · Zbl 1213.65148 · doi:10.1155/2011/724927
[9] Simos, T. E., New stable closed Newton-Cotes trigonometrically fitted formulae for long-time integration, Abstract and Applied Analysis (2012) · Zbl 1242.65026 · doi:10.1155/2012/182536
[10] Simos, T. E., Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag, Journal of Applied Mathematics (2012) · Zbl 1247.65096 · doi:10.1155/2012/420387
[11] Anastassi, Z. A.; Simos, T. E., A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems, Journal of Computational and Applied Mathematics, 236, 16, 3880-3889 (2012) · Zbl 1246.65105 · doi:10.1016/j.cam.2012.03.016
[12] Akram, G.; Rehman, H. U., Solution of first order singularly perturbed initial value problem in reproducing kernel Hilbert space, European Journal of Scientific Research, 53, 4, 516-523 (2011)
[13] Akram, G.; Rehman, H. U., Numerical solution of eighth order boundary value problems in reproducing kernel space, Numerical Algorithms, 62, 3, 527-540 (2013) · Zbl 1281.65101 · doi:10.1007/s11075-012-9608-4
[14] Akram, G.; Rehman, H. U., Solution of fifth order boundary value problems in the reproducing kernel space, Middle East Journal of Scientific Research, 10, 2, 191-195 (2011)
[15] Yao, H., New algorithm for the numerical solution of the integro-differential equation with an integral boundary condition, Journal of Mathematical Chemistry, 47, 3, 1054-1067 (2010) · Zbl 1187.92094 · doi:10.1007/s10910-009-9628-z
[16] Pandey, P. K., High order finite difference method for numerical solution of general two- point boundary value problems involving sixth order differential equation, International Journal of Pure and Applied Mathematics, 76, 3, 317-326 (2012) · Zbl 1250.65102
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.