Extended Fisher-Kolmogorov Denklemini Çözmek İçin Kuartik B-Spline Diferansiyel Kuadratur Metot

Year 2019, , 56 - 62, 24.03.2019

Ali Başhan

https://doi.org/10.18185/erzifbed.416702

Cited By: 2

Abstract

Extended Fisher-Kolmogorov (EFK) denkleminin bazı çözümleri kuartik B-spline diferansiyel quadrature metot (DQM) ile elde edildi. İkinci mertebeden ağırlık katsayıları kuartik B-spline fonksiyonlar ile direkt olarak elde edildi. Kuartik B-spline fonksiyonların dördüncü mertebeden türevleri mevcut olmadığından, dördüncü mertebeden ağırlık katsayıları matris çarpımı yaklaşımı ile elde edildi. EFK denklemi DQM ile ayrıklaştırıldıktan sonra adi diferansiyel denklem sistemi elde edildi ve kararlılığı güçlü bir şekilde koruyan Runge-Kutta metot ile zamana bağlı integre edildi. Metodun tamlığını kontrol etmek için üç adet test problemi çözüldü ve L₂ile L_∞hata normları hesaplandı.

Keywords

Kısmi diferansiyel denklemler, Diferansiyel Quadrature Metot, Runge-Kutta metot

Quartic B-spline Differential Quadrature Method for Solving the Extended Fisher-Kolmogorov Equation

Abstract

Some numerical solutions of the extended Fisher-Kolmogorov(EFK) equation have been obtained via quartic B-spline differential quadrature method(DQM). 2^ndorder weighting coefficients are obtained directly by quartic B-splines. Since the 4^thorder derivatives of quartic B-splines do not exist, the 4^thorder weighting coefficients have been obtained by matrix multiplication approach. After the discretization of the eFK equation via DQM, ordinary differential equation systems have been obtained and strong stability preserving Runge-Kutta method has been used for time integration. To be able to control the accuracy of the method three test problems have been solved and error norms L₂and L_∞are calculated.

Keywords

Partial Differential Equations, Differential Quadrature Method, Runge-Kutta method

References

• Başhan A, Karakoç, SBG, Geyikli T. Approximation of the KdVB equation by the quintic B-spline differential quadrature method, Kuwait Journal of Science, 2015; 42; 2; 67-92.Başhan A, Uçar Y, Yağmurlu NM, Esen A. Numerical Solution of the Complex Modified Korteweg-de Vries Equation by DQM, Journal of Physics:Conference Series 2016; 766; 012028 doi:10.1088/1742-6596/766/1/012028Başhan A, Yağmurlu NM, Uçar Y, Esen A. An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method, Chaos, Solitons and Fractals, 2017; 100; 45-56.Başhan A. An effective application of differential quadrature method based on modified cubic B-splines to numerical solutions of KdV equation, Turkish Journal of Mathematics, (2018) 42: 373-394.Başhan A, Uçar Y, Yağmurlu NM, Esen A. A new perspective for quintic B-spline based Crank-Nicolson differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation, Eur. Phys. J. Plus (2018) 133: 12Bellman R, Kashef BG, Casti J. Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, 1972;10; 40-52.Coullet P, Elphick C, Repaux D. Nature of spatial chaos,Phys. Rev. Lett. 1987; 58;5; 431-434.Danumjaya P, Pani AK. Finite element methods for the extended Fisher-Kolmogorov equation. Journal of Computational and Applied Mathematics, 2005; 174; 101-117.Dee GT, van Saarloos W. Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 1988; 60; 25; 2641-2644.Hornreich RM, Luban M, Shtrikman S. Critical behaviour at the onset of k-space instability at the line, Phys.Rev. Lett. 1975; 35; 1678-1681.Ketcheson DI. Runge–Kutta methods with minimum storage implementations, Journal of Computational Physics, 2010; 229; 1763–1773.Korkmaz A, Dağ I. Shock wave simulations using Sinc Differential Quadrature Method, International Journal for Computer-Aided Engineering and Software, 2011; 28; 6; 654-674.Mittal RC, Arora G. Quintic B-spline collocation method for numerical solution of the extended Fisher-Kolmogorov equation, Int. J. Appl. Math Mech. 2010; 6; 1; 74-85.Mittal RC, Dahiya S. A study of quintic B-spline based differential quadrature method for a class of semi-linear Fisher-Kolmogorov equations, Alexandria Engineering Journal, 2016; 55; 2893-2899.Prenter P M. Splines and Variational Methods, New York: John Wiley, 1975.Shu, C. Differential Quadrature and its application in engineering, Springer- Verlag London Ltd., 2000.Shu C, Xue H. Explicit computation of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration, 1997; 204; 3; 549-555.Shu C, Wu YL. Integrated radial basis functions-based differential quadrature method and its performance, Int. J. Numer. Meth. Fluids, 2007; 53; 969-984.Striz AG, Wang X, Bert CW. Harmonic differential quadrature method and applications to analysis of structural components”, Acta Mechanica, 1995; 111; 85-94.van Saarloos W. Dynamical velocity selection: marginal stability, Phys. Rev. Lett. 1987; 58; 24; 2571-2574.van Saarloos W. Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection, Phys. Rev. Lett. A 1988; 37;1;211-229.van Saarloos W. Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence, Phys. Rev. Lett. A 1989; 39; 12; 6367-6389.Zhu G. Experiments on director waves in nematic liquid crystals, Phys. Rev. Lett. 1982; 49; 1332-1335.Zimmerman W. Propagating fronts near a Lifschitz point, Phys.Rev. Lett. 1991; 66; 1546.