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Kombine KdV-mKdV Denkleminin Kuintik B-Splayn Diferansiyel Kuadratür Yöntemiyle Sayısal Yaklaşımlar

Year 2019, Volume: 9 Issue: 2, 386 - 403, 30.12.2019
https://doi.org/10.37094/adyujsci.526264

Abstract

Bu makalede, kombine Korteweg-de Vries ve modifiye edilmiş Korteweg-de Vries denkleminin (kombine KdV-mKdV) sayısal yaklaşımını elde etmek için kuintik B-spline diferansiyel kuadratür yöntemi (QBDQM) uygulanmıştır. Yöntemin etkinliği ve doğruluğu, maksimum hata normu L_\infinity ve ayrık kök ortalama kare hatası L_2 hesaplanarak ölçülmüştür. Yeni elde edilen sayısal sonuçlar, yayınlanan sayısal sonuçlarla karşılaştırıldı ve karşılaştırma, yöntemin, kombine KdV-mKdV denklemini çözmek için etkili bir sayısal şema olduğunu göstermiştir. Aynı zamanda bir kararlılık analizi de yapılmıştır.

References

  • [1] Wadati, M., Wave propagation in nonlinear lattice I, Journal of the Physical Soceity of Japan, 38(3), 673-680, 1975.
  • [2] Wadati, M., Wave propagation in nonlinear lattice II, Journal of the Physical Soceity of Japan, 38(3), 681-686, 1975.
  • [3] Taha, T.R., Inverse scattering transform numerical schemes for nonlinear evolution equations and the method of lines, Applied Numerical Mathematics, 20, 181-187, 1996.
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  • [7] Lu, D., Shi, Q., New solitary wave solutions for the combined KdV-MKdV equation, Journal of Information & Computational Science, 7, 1733-1737, 2010.
  • [8] Bellman, R., Kashef, B. G., Casti, J., Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, Vol. 10, 40-52, 1972.
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  • [20] Striz, A.G., Wang, X., Bert, C. W., Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica, 111, 85-94, 1995.
  • [21] Korkmaz, A., Dağ, I., Shock wave simulations using Sinc differential quadrature method, International Journal for Computer-Aided Engineering and Software, 28(6), 654-674, 2011.
  • [22] Bonzani, I., Solution of non-linear evolution problems by parallelized collocation-interpolation methods, Computers & Mathematics and Applications, 34(12), 71-79, 1997.
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  • [27] Korkmaz, A., Numerical solutions of some partial differential equations using B-spline differential quadrature methods, PhD, Eskişehir Osmangazi University, Eskişehir, 2010.
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  • [29] Ketcheson, D.I., Runge–Kutta methods with minimum storage implementations, Journal of Computational Physics, 229, 1763–1773, 2010.
  • [30] Ketcheson, D.I., Highly efficient strong stability preserving Runge-Kutta methods with Low-Storage Implementations, SIAM Journal on Scientific Computing, 30(4), 2113–2136, 2008.

Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method

Year 2019, Volume: 9 Issue: 2, 386 - 403, 30.12.2019
https://doi.org/10.37094/adyujsci.526264

Abstract

In this paper, quintic B-spline differential quadrature method (QBDQM) has been used to obtain the numerical approximation of the combined Korteweg-de Vries and modified Korteweg-de Vries equation (combined KdV-mKdV). The efficiency and effectiveness of the proposed method has been tested by computing the maximum error norm L_\infinity and discrete root mean square error L_2. The newly found numerical approximations have been compared to available numerical approximations and this comparison has shown that the proposed method is an efficient one for solving 

References

  • [1] Wadati, M., Wave propagation in nonlinear lattice I, Journal of the Physical Soceity of Japan, 38(3), 673-680, 1975.
  • [2] Wadati, M., Wave propagation in nonlinear lattice II, Journal of the Physical Soceity of Japan, 38(3), 681-686, 1975.
  • [3] Taha, T.R., Inverse scattering transform numerical schemes for nonlinear evolution equations and the method of lines, Applied Numerical Mathematics, 20, 181-187, 1996.
  • [4] Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Soliton Fract, 16, 819–39, 2003.
  • [5] Peng, Y., New exact solutions to the combined KdV and mKdV equation, International Journal Theoretical Physics, 42(4), 863–868, 2003.
  • [6] Bekir, A., On traveling wave solutions to combined KdV–mKdV equation and modified Burgers–KdV equation, Communications in Nonlinear Science and Numerical Simulation, 14, 1038–1042, 2009.
  • [7] Lu, D., Shi, Q., New solitary wave solutions for the combined KdV-MKdV equation, Journal of Information & Computational Science, 7, 1733-1737, 2010.
  • [8] Bellman, R., Kashef, B. G., Casti, J., Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations, Journal of Computational Physics, Vol. 10, 40-52, 1972.
  • [9] Shu, C., Differential Quadrature and Its Application in Engineering, Springer-Veralg London Ltd., London, 2000.
  • [10] Bellman, R., Kashef, B., Lee, E.S., Vasudevan, R., Differential quadrature and splines, Computers and Mathematics with Applications, Pergamon, Oxford, 1(3-4), 371-376, 1975.
  • [11] Quan, J.R., Chang, C.T., New sightings in involving distributed system equations by the quadrature methods-I, Computers and Chemical Engineering, 13, 779-88, 1989.
  • [12] Quan, J.R., Chang, C.T., New sightings in involving distributed system equations by the quadrature methods-II, Computers and Chemical Engineering, 13, 1017-1024, 1989.
  • [13] Shu, C., Richards, B.E., Application of generalized differential quadrature to solve two dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 15, 791-798, 1992.
  • [14] Shu, C., Xue, H., Explicit computation of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration, 204(3), 549-55, 1997.
  • [15] Zhong, H., Spline-based differential quadrature for fourth order equations and its application to Kirchhoff plates, Applied Mathematical Modelling, 28, 353-66, 2004.
  • [16] Guo, Q. and Zhong, H., Non-linear vibration analysis of beams by a spline-based differential quadrature method, Journal of Sound and Vibration, 269, 413-420, 2004.
  • [17] Zhong, H., Lan, M., Solution of nonlinear initial-value problems by the spline-based differential quadrature method, Journal of Sound and Vibration, 296, 908-918, 2006. [18] Cheng, J., Wang, B., Du, S., A theoretical analysis of piezoelectric/composite laminate with larger-amplitude deflection effect, Part II: Hermite differential quadrature method and application, International Journal of Solids and Structures, 42, 6181-6201, 2005.
  • [19] Shu, C., Wu, Y.L., Integrated radial basis functions-based differential quadrature method and its performance, The International Journal for Numerical Methods in Fluids, 53, 969-84, 2007.
  • [20] Striz, A.G., Wang, X., Bert, C. W., Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica, 111, 85-94, 1995.
  • [21] Korkmaz, A., Dağ, I., Shock wave simulations using Sinc differential quadrature method, International Journal for Computer-Aided Engineering and Software, 28(6), 654-674, 2011.
  • [22] Bonzani, I., Solution of non-linear evolution problems by parallelized collocation-interpolation methods, Computers & Mathematics and Applications, 34(12), 71-79, 1997.
  • [23] Başhan, A., Numerical solutions of some partial differential equations with B-spline differential quadrature method, PhD, İnönü University, Malatya, 2015.
  • [24] Korkmaz, A., Dağ, I., Cubic B-spline differential quadrature methods for the advection-diffusion equation, International Journal of Numerical Methods for Heat & Fluid Flow, 22(8), 1021-1036, 2012.
  • [25] Korkmaz, A., Dağ, I., Numerical simulations of boundary-forced RLW equation with cubic B-Spline-based differential quadrature methods, Arabian Journal for Science and Engineering., 38(5), 1151-1160, 2013.
  • [26] Korkmaz, A., Dağ, I., Cubic B-spline differential quadrature methods and stability for Burgers’ equation, International Journal for Computer-Aided Engineering and Software, 30(3), 320-344, 2013.
  • [27] Korkmaz, A., Numerical solutions of some partial differential equations using B-spline differential quadrature methods, PhD, Eskişehir Osmangazi University, Eskişehir, 2010.
  • [28] Prenter, P.M., Splines and variational methods, John Wiley Publications, New York, 1975.
  • [29] Ketcheson, D.I., Runge–Kutta methods with minimum storage implementations, Journal of Computational Physics, 229, 1763–1773, 2010.
  • [30] Ketcheson, D.I., Highly efficient strong stability preserving Runge-Kutta methods with Low-Storage Implementations, SIAM Journal on Scientific Computing, 30(4), 2113–2136, 2008.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Murat Yağmurlu 0000-0003-1593-0254

Yusuf Uçar 0000-0003-1469-5002

Ali Başhan 0000-0001-8500-493X

Publication Date December 30, 2019
Submission Date February 13, 2019
Acceptance Date December 18, 2019
Published in Issue Year 2019 Volume: 9 Issue: 2

Cite

APA Yağmurlu, M., Uçar, Y., & Başhan, A. (2019). Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method. Adıyaman University Journal of Science, 9(2), 386-403. https://doi.org/10.37094/adyujsci.526264
AMA Yağmurlu M, Uçar Y, Başhan A. Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method. ADYU J SCI. December 2019;9(2):386-403. doi:10.37094/adyujsci.526264
Chicago Yağmurlu, Murat, Yusuf Uçar, and Ali Başhan. “Numerical Approximation of the Combined KdV-MKdV Equation via the Quintic B-Spline Differential Quadrature Method”. Adıyaman University Journal of Science 9, no. 2 (December 2019): 386-403. https://doi.org/10.37094/adyujsci.526264.
EndNote Yağmurlu M, Uçar Y, Başhan A (December 1, 2019) Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method. Adıyaman University Journal of Science 9 2 386–403.
IEEE M. Yağmurlu, Y. Uçar, and A. Başhan, “Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method”, ADYU J SCI, vol. 9, no. 2, pp. 386–403, 2019, doi: 10.37094/adyujsci.526264.
ISNAD Yağmurlu, Murat et al. “Numerical Approximation of the Combined KdV-MKdV Equation via the Quintic B-Spline Differential Quadrature Method”. Adıyaman University Journal of Science 9/2 (December 2019), 386-403. https://doi.org/10.37094/adyujsci.526264.
JAMA Yağmurlu M, Uçar Y, Başhan A. Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method. ADYU J SCI. 2019;9:386–403.
MLA Yağmurlu, Murat et al. “Numerical Approximation of the Combined KdV-MKdV Equation via the Quintic B-Spline Differential Quadrature Method”. Adıyaman University Journal of Science, vol. 9, no. 2, 2019, pp. 386-03, doi:10.37094/adyujsci.526264.
Vancouver Yağmurlu M, Uçar Y, Başhan A. Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method. ADYU J SCI. 2019;9(2):386-403.

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