Birleştirilmiş KdV-mKdV Denkleminin Yaklaşık Analitik Çözümleri Üzerine Bir Çalışma

Yıl 2020, Cilt: 10 Sayı: 4, 917 - 924, 15.10.2020

Turgut Ak , Bilge Inan

https://doi.org/10.17714/gumusfenbil.678682

Öz

Bu çalışmada, araştırma indirgenmiş diferansiyel dönüşüm metodunu (İDDM)
kullanarak birleştirilmiş KdV-mKdV denkleminin solitary dalga çözümleri üzerine
odaklanmaktadır. Önerilen metodun geçerliliğini kanıtlamak için, mutlak hata
vasıtasıyla yaklaşık analitik çözümler ve tam çözümler karşılaştırılmıştır.
Elde edilen sonuçlar grafiklerle temsil edilmiştir. Zaman ve dispersiyon
parametresinin



















 analitik çözümler üzerindeki
etkileri araştırılmıştır. Sonuç olarak, uygulanan yöntemin benzer tipteki
denklemler için oldukça hassas ve başarılı olduğu söylenebilir.

Anahtar Kelimeler

Birleştirilmiş KdV-mKdV Denklemi, Gardner Denklemi, İDDM, İndirgenmiş Diferansiyel Dönüşüm Metodu, Solitary Dalga

A Study on Approximate Analytic Solutions of the Combined KdV-mKdV Equation

Öz

In this paper, the investigation
focuses on solitary wave solutions of the combined KdV-mKdV equation by using
reduced differential transform method (RDTM). To prove validity of the proposed
method, the approximate analytic solutions and exact solutions of the equation
are compared via absolute errors. The obtained results are represented by
graphics. The effects of the time and dispersion parameter



















 on analytic solutions are investigated. As a
result, it can be said that the applied method is quite precise and successful
for similar type equations.

Anahtar Kelimeler

Combined KdV-mKdV Equation, Gardner Equation, RDTM, Reduced Differential Transform Method, Solitary Wave

Kaynakça

• Abazari, R. and Abazari, M., 2013. Numerical Study of Burger-Huxley Equations via Reduced Differential Transform Method. Computational and Applied Mathematics, 32(1), 1-17.
• Abazari, R. and Soltanalizadeh, B., 2013. Reduced Differential Transform Method and Its Application on Kawahara Equations. Thai Journal of Mathematics, 11(1), 199-216.
• Ak, T., Triki, H., Dhawan, S. and Erduran, K.S., 2018. Theoretical and Numerical Investigations on Solitary Wave Solutions of Gardner Equation. The European Physical Journal Plus, 133, 382.
• Ak, T., 2019. Numerical Experiments for Long Nonlinear İnternal Waves via Gardner Eqaution with Dual-Power Law Nonlinearity. International Journal of Modern Physics C, 30(9), 1950066.
• Al-Amr, M.O., 2014. New Applications of Reduced Differential Transform Method. Alexandria Engineering Journal, 53(1), 243-247.
• Biswas, A. and Zerrad, E., 2008. Soliton Perturbation Theory for the Gardner Equation. Advanced Studies in Theoretical Physics, 2(16), 787-794.
• Hamdi, S., Morse, B., Halphen, B. and Schiesser, W., 2011. Analytical Solutions of Long Nonlinear Internal Waves: Part I. Natural Hazards, 57(3), 597-607.
• Hesam, S., Nazemi, A. and Haghbin, A., 2012. Reduced Differential Transform Method for Solving the Fornberg-Whitham Type Equation. International Journal of Nonlinear Science, 13(2), 158-162.
• Hirota, R., 1980. Direct Methods in Soliton Theory. In: Bullough R.K., Caudrey P.J. (eds) Solitons. Topics in Current Physics, vol 17. Springer, Berlin, Heidelberg
• Hoover, J.R., Grant, W. J., 1983. Numerical Fitting of the Gardner Equation to Hydraulic Conductivity and Water Retention Data. Transactions of the ASAE, 26(5), 1401-1408.
• Kaya, D. and Inan, I.E., 2005. A Numerical Application of the Decomposition Method for the Combined KdV-mKdV Equation. Applied Mathematics and Computation, 168(2), 915-926.
• Keskin, Y. and Oturanc, G., 2009. Reduced Differential Transform Method for Partial Differential Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 106, 741-749.
• Keskin, Y., 2010. Numerical Solution of Regularized Long Wave Equation by Reduced Differential Transform Method. Applied Mathematical Sciences, 4(25), 1221-1231.
• Keskin, Y. and Oturanc, G., 2010a. Reduced Differential Transform Method for Generalized KdV Equations. Mathematical and Computational Applications, 15(3), 382-393.
• Keskin, Y. and Oturanc, G., 2010b. Reduced Differential Transform Method for Solving Linear and Nonlinear Wave Equations. Iranian Journal of Science & Technology, Transaction A, 34(2), 113-122.
• Lou, S.-Y., Chen, L.-L., 1994. Solitary Wave Solutions and Cnoidal Wave Solutions to the Combined KdV and mKdV Equation. Mathematical Methods in the Applied Sciences, 17(5), 339-347.
• Lu, D. and Shi, Q., 2010. New Jacobi Elliptic Functions Solutions for the Combined KdV-MKdV Equation. International Journal of Nonlinear Science, 10(3), 320-325.
• Mohamad, M.N.B., 1992. Exact Solutions to the Combined KdV and mKdV Equation. Mathematical Methods in the Applied Sciences, 15(2), 73-78.
• Nakoulima, O., Zahibo, N., Pelinovsky, E., Talipova, T., Slunyaev A. and Kurkin A., 2004. Analytical and Numerical Studies of the Variable-Coefficient Gardner Equation. Applied Mathematics and Computation, 152(2), 449-471.
• Saravanan, A. and Magesh, N., 2013. A Comparison Between the Reduced Differential Transform Method and the Adomian Decomposition Method for the Newell-Whitehead-Segel Equation. Journal of the Egyptian Mathematical Society, 21(3), 259-265.
• Triki, H., Taha, T.R. and Wazwaz, A.-M., 2010. Solitary Wave Solutions for A Generalized KdV-mKdV Equation with Variable Coefficients. Mathematics and Computers in Simulation, 80(9), 1867-1873.
• Wazwaz, A.-M., 2007. New Solitons and Kink Solutions for the Gardner Equation. Communications in Nonlinear Science and Numerical Simulation, 12(8), 1395-1404.
• Zhang, J., 1998. New Solitary Wave Solution of the Combined KdV and mKdV Equation. International Journal of Theoretical Physics, 37(5), 1541-1546.
• Zhang, J., Wu, F., Shi, J., 2000. Simple Soliton Solution Method for the Combined KdV and mKdV Equation. International Journal of Theoretical Physics, 39(6), 1697-1702.