A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline

Yıl 2024, Cilt: 5 Sayı: 2, 143 – 158, 31.07.2024

İhsan Çelikkaya

https://doi.org/10.54974/fcmathsci.1398394

Öz

Kaynakça

• Abbasbandy S., Solitary wave solutions to the modified form of Camassa–Holm equation by means of the homotopy analysis method, Chaos, Solitons and Fractals, 39, 428-435, 2009.
• Ali I., Haq S., Hussain M., Nisar K.S., Arifeen S.U., On the analysis and application of a spectral collocation scheme for the nonlinear two-dimensional fractional diffusion equation, Results in Physics, 56, 1-11, 2024.
• Behera R., Mehra M., Approximate solution of modified Camassa–Holm and modified Degasperis– Procesi equations using wavelet optimized finite difference method, International Journal of Wavelets, Multiresolution and Information Processing, 11(2), 1-13, 2013.
• Camassa R., Holm D.D., An integrable shallow water equation with peaked solitions, Physical Review Letters, 71, 1661-1664, 1993.
• Degasperis A., Procesi M., Asymptotic Integrability. In: Degasperis A., et al., Eds., Symmetry and Perturbation Theory, World Scientific, 23-37, 1999.
• Ganji D.D., Sadeghi E.M.M., Rahmat M.G., Modified forms of Degaperis-Procesi and Camassa-Holm equations solved by Adomian’s decomposition method and comparison with HPM and exact solution, Acta Applicandae Mathematicae, 104, 303-311, 2008.
• Keskin P., Trigonometric B-spline solutions of the RLW equation, Ph D., Eskişehir Osmangazi University, Eskişehir, Türkiye, 2017.
• Liu Z., Ouyang Z., A note on solitary waves for modified forms of Camassa–Holm and Degasperis– Procesi equations, Physics Letters A, 366(4-5), 377-381, 2007.
• Logan D.L., A First Course in the Finite Element Method (Fourth Edition), Thomson, 2007.
• Lundmark H., Szmigielski J., Multi-peakon solutions of the Degasperis–Procesi equation, Inverse Problems, 19(6), 1241-1245, 2003.
• Manafian J., Shahabi R., Asaspour M., Zamanpour I., Jalali J., Construction of exact solutions to the modified forms of DP and CH equations by analytical methods, Statistics, Optimization and Information Computing, 3, 336-347, 2015.
• Mustafa O.G., A note on the Degasperis–Procesi equation, Journal of Mathematical Physics, 12(1), 10-14, 2005.
• Rao Singiresu S., The Finite Element Method in Engineering (Fifth Edition), Elsevier, 2011.
• VonNeumann J., Richtmyer R.D., A method for the numerical calculation of hydrodynamic shocks, Journal of Applied Physics, 21, 232-237, 1950.
• Walz G., Identities for trigonometric B-splines with an application to curve design, BIT Numerical Mathematics, 37, 189-201, 1997.
• Wasim I., Abbas M., Iqbal M.K., Numerical solution of modified forms of Camassa-Holm and Degasperis-Procesi equations via quartic B-spline collocation method, Communications in Mathematics and Applications, 9(3), 393-409, 2018.
• Wang Q., Tang M., New exact solutions for two nonlinear equations, Physics Letters A, 372(17), 2995-3000, 2008.
• Wazwaz A.M., New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa– Holm equations, Applied Mathematics and Computation, 186, 130-141, 2007.
• Wazwaz A.M., Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations, Physics Letters A, 352(6), 500-504, 2006.
• Yousif M.A., Mahmood B.A., Easif F.H., A new analytical study of modified Camassa-Holm and Degaperis-Procesi equations, American Journal of Computational Mathematics, 5, 267-273, 2015.
• Yusufoglu E., New solitonary solutions for modified forms of DP and CH equations using Exp-function method, Chaos, Solitons and Fractals, 39(5), 2442-2447, 2009.
• Zada L., Nawaz R., An Efficient approach for solution of Camassa-Holm and Degasperis-Procesl equations, UPB Scientific Bulletin, Series D: Mechanical Engineering, 79(3), 25-34, 2017.
• Zhang B.G., Li S.Y., Liu Z.R., Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations, Physics Letters A, 372(11), 1867-1872, 2008.
• Zhang B.G., Liu Z.R., Mao J.F., New exact solutions for mCH and mDP equations by auxiliary equation method, Applied Mathematics and Computation, 217, 1306-1314, 2010.
• Zhou Y., Blow-up phenomenon for the integrable Degasperis–Procesi equation, Physics Letters A, 328(2-3), 157-162, 2004.

A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline

Yıl 2024, Cilt: 5 Sayı: 2, 143 – 158, 31.07.2024

İhsan Çelikkaya

https://doi.org/10.54974/fcmathsci.1398394

Öz

In this study, the soliton solutions of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations, called modified b-equations with important physical properties, were obtained. The soliton waves’ movement and positions formed by solving the mCH and mDP equations were calculated. Ordinary differential equation systems were obtained using trigonometric quintic B-spline bases for position and time direction derivatives in the equations to obtain numerical solutions. An algebraic equation system was then created by writing Crank-Nicolson type approximations for time and position-dependent terms. The stability analysis of this system was examined using the von-Neumann Fourier series method. L₂, L_{∞} and absolute error norms were used to measure the numerical results’ convergence to the real solution. The numerical results calculated were compared with the exact solution and some studies in the literature.

Anahtar Kelimeler

Modified Camassa-Holm, Modified Degasperis-Procesi Equation, Soliton Waves, Trigonometric Quintic B-spline bases, Collocation Method

Kaynakça

• Abbasbandy S., Solitary wave solutions to the modified form of Camassa–Holm equation by means of the homotopy analysis method, Chaos, Solitons and Fractals, 39, 428-435, 2009.
• Ali I., Haq S., Hussain M., Nisar K.S., Arifeen S.U., On the analysis and application of a spectral collocation scheme for the nonlinear two-dimensional fractional diffusion equation, Results in Physics, 56, 1-11, 2024.
• Behera R., Mehra M., Approximate solution of modified Camassa–Holm and modified Degasperis– Procesi equations using wavelet optimized finite difference method, International Journal of Wavelets, Multiresolution and Information Processing, 11(2), 1-13, 2013.
• Camassa R., Holm D.D., An integrable shallow water equation with peaked solitions, Physical Review Letters, 71, 1661-1664, 1993.
• Degasperis A., Procesi M., Asymptotic Integrability. In: Degasperis A., et al., Eds., Symmetry and Perturbation Theory, World Scientific, 23-37, 1999.
• Ganji D.D., Sadeghi E.M.M., Rahmat M.G., Modified forms of Degaperis-Procesi and Camassa-Holm equations solved by Adomian’s decomposition method and comparison with HPM and exact solution, Acta Applicandae Mathematicae, 104, 303-311, 2008.
• Keskin P., Trigonometric B-spline solutions of the RLW equation, Ph D., Eskişehir Osmangazi University, Eskişehir, Türkiye, 2017.
• Liu Z., Ouyang Z., A note on solitary waves for modified forms of Camassa–Holm and Degasperis– Procesi equations, Physics Letters A, 366(4-5), 377-381, 2007.
• Logan D.L., A First Course in the Finite Element Method (Fourth Edition), Thomson, 2007.
• Lundmark H., Szmigielski J., Multi-peakon solutions of the Degasperis–Procesi equation, Inverse Problems, 19(6), 1241-1245, 2003.
• Manafian J., Shahabi R., Asaspour M., Zamanpour I., Jalali J., Construction of exact solutions to the modified forms of DP and CH equations by analytical methods, Statistics, Optimization and Information Computing, 3, 336-347, 2015.
• Mustafa O.G., A note on the Degasperis–Procesi equation, Journal of Mathematical Physics, 12(1), 10-14, 2005.
• Rao Singiresu S., The Finite Element Method in Engineering (Fifth Edition), Elsevier, 2011.
• VonNeumann J., Richtmyer R.D., A method for the numerical calculation of hydrodynamic shocks, Journal of Applied Physics, 21, 232-237, 1950.
• Walz G., Identities for trigonometric B-splines with an application to curve design, BIT Numerical Mathematics, 37, 189-201, 1997.
• Wasim I., Abbas M., Iqbal M.K., Numerical solution of modified forms of Camassa-Holm and Degasperis-Procesi equations via quartic B-spline collocation method, Communications in Mathematics and Applications, 9(3), 393-409, 2018.
• Wang Q., Tang M., New exact solutions for two nonlinear equations, Physics Letters A, 372(17), 2995-3000, 2008.
• Wazwaz A.M., New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa– Holm equations, Applied Mathematics and Computation, 186, 130-141, 2007.
• Wazwaz A.M., Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations, Physics Letters A, 352(6), 500-504, 2006.
• Yousif M.A., Mahmood B.A., Easif F.H., A new analytical study of modified Camassa-Holm and Degaperis-Procesi equations, American Journal of Computational Mathematics, 5, 267-273, 2015.
• Yusufoglu E., New solitonary solutions for modified forms of DP and CH equations using Exp-function method, Chaos, Solitons and Fractals, 39(5), 2442-2447, 2009.
• Zada L., Nawaz R., An Efficient approach for solution of Camassa-Holm and Degasperis-Procesl equations, UPB Scientific Bulletin, Series D: Mechanical Engineering, 79(3), 25-34, 2017.
• Zhang B.G., Li S.Y., Liu Z.R., Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations, Physics Letters A, 372(11), 1867-1872, 2008.
• Zhang B.G., Liu Z.R., Mao J.F., New exact solutions for mCH and mDP equations by auxiliary equation method, Applied Mathematics and Computation, 217, 1306-1314, 2010.
• Zhou Y., Blow-up phenomenon for the integrable Degasperis–Procesi equation, Physics Letters A, 328(2-3), 157-162, 2004.

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Ayrıntılar

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