Approximate Solutions of Singularly Perturbed Nonlinear Ill-posed and Sixth-order Boussinesq Equations with Hybrid Method

Year 2019, Volume: 8 Issue: 2, 451 - 458, 28.06.2019

Derya Arslan

https://doi.org/10.17798/bitlisfen.491847

Cited By: 3

Abstract

The aim of this paper is to obtain the approximate solution of singularly perturbed ill-posed and sixth-order
Boussinesq equation by hybrid method (differential transform and finite difference method) as a different
alternative method. Differential transform method is applied for 𝑡 −time variable and the finite difference method
(central difference approach) is applied for 𝑥 −position variable. Two examples are presented to demonstrate the
efficiency and reliability of the hybrid method. Numerical results are given and compared with exact solution and
in literature RDTM solution. The numerical data show that hybrid method is a powerful, quite efficient and is
practically well suited for solving nonlinear singular perturbed Boussinesq equations.

Keywords

Sixth-order Boussinesq Equation, Differential Transform Method, Finite Difference Method, Approximate Solution

Hibrit Metot ile Singüler Pertürbe Nonlineer Ill-posed ve Altıncı Mertebe Boussinesq Denklemlerinin Yaklaşık Çözümleri

Abstract

Bu çalışmanın amacı, singüler pertürbe lineer olmayan ill-posed ve altıncı mertebeden Boussinesq denkleminin
farklı bir alternatif yöntem olan hibrit metotla (diferansiyel dönüşüm ve sonlu fark metodu) yaklaşık çözümünü
elde etmektir. 𝑡 −zaman değişkeni için diferansiyel dönüşüm metodu ve 𝑥 −konum değişkeni için sonlu fark
metodu (merkezi fark yaklaşımı) uygulanmıştır. Hibrit yöntemin etkinliğini ve güvenilirliğini göstermek için iki
örnek sunulmuştur. Nümerik sonuçlar, kesin çözüm ve literatürde yer alan RDTM çözümü ile karşılaştırılmıştır.
Sayısal veriler bu yöntemin güçlü, oldukça etkili olduğunu ve nonlineer singüler pertürbe Boussinesq
denklemlerini çözmek için pratik olarak uygun olduğunu göstermektedir.

Keywords

Altıncı Mertebe Boussinesq Denklemi, Ill-posed Boussinesq Denklemi, Diferansiyel Dönüşüm Metodu, Sonlu Fark Metodu, Yaklaşık Çözüm

References

• 1. Dash R.K., Daripa, P. 2002. Analytical and Numerical Studies of Singularly Perturbed Boussinesq Equation, Applied Mathematics and Computation, 126: 1–30. 2. Daripa, P., Dash, R.K. 2001. Weakly Non-local Solitary Wave Solutions of a Singularly Perturbed Boussinesq Equation, Mathematics and Computers in Simulation, 55: 393– 405.3. Song, C., Li, H., Li, J. 2013. Initial Boundary Value Problem for the Singularly Perturbed Boussinesq-Type Equation, Discrete and Continuous Dynamical Systems, 2013: 709–717. 4. Zhou, J.K. 1986. Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China.5. Darapi, P., Hua, W. 1999. A Numerical Method for Solving an ill-posed Boussinesq Equation Arising in Water Waves and Nonlinear Lattices. Applied Mathematics and Computation, 101, 159–207.6. Süngü, İ., Demir, H. 2012. Solutions of the System of Differential Equations by Differential Transform / Finite Difference Method, Nwsa-Physical Sciences, 7: 1308-7304.7. Süngü, İ., Demir, H. 2012. Application of the Hybrid Differential Transform Method to the Nonlinear Equations, Applied Mathematics, 3: 246-250.8. Yeh, Y.L., Wang, C.C., Jang, M.J. 2007. Using Finite Difference and Differential Transformation Method to Analyze of Large Deflections of Orthotropic Rectangular Plate Problem, Applied Mathematics and Computation, 190:1146-1156.9. Chu, H.P., Chen, C.L. 2008. Hybrid Differential Transform and Finite Difference Method to Solve the Nonlinear Heat Conduction Problem, Communication in Nonlinear Science and Numerical Simulation, 13: 1605-1614.10. Chu, S.P. 2014. Hybrid Differential Transform and Finite Difference Method to Solve the Nonlinear Heat Conduction Problems, WHAMPOA - An Inter disciplinary Journal, 66: 15-26.11. Chen, C.K., Lai, H-Y., Liu, C.C. 2009. Nonlinear Micro Circular Plate Analysis Using Hybrid Differential Transformation / Finite Difference Method, CMES, 40: 155-174.12. Song, C.,Li, J., Gao, R. 2014. Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation, Journal of Applied Mathematics, 2014: 7 pages.13. Ayaz, F. 2004. Solution of the System of Differential Equations by Differential Transform Method, Appl. Math. Comput., 147: 547-567.14. Ayaz, F. 2004. Applications of Differential Transform Method to Differential Algebraic Equations, Appl. Math. Comput., 152: 649-657.15. Mohyud-Din, S.T., Muhammad Aslam, N. 2009. Homotopy Perturbation Method for Solving Partial Differential Equations, Zeitschrift Naturforschung, 64, 157-170.16. Feng, Z. 2003. Traveling Solitary Wave Solutions to the Generalized Boussinesq Equation, 37: 17–23.17. Cakır, M., Arslan, D. 2016. Reduced Differential Transform Method for Singularly Perturbed Sixth-Order Boussinesq Equation, Mathematics and Statistics: Open Access, 2(2): MSOA-2-014.18. Arslan, D. 2018. Differential Transform Method for Singularly Perturbed Singular Differential Equations, Journal of the Institute of Natural &Applied Sciences, 23 (3): 254-260.19. Arslan, D. 2018. The Approximate Solution of Fokker-Planck Equation with Reduced Differential Transform Method, Erzincan UniversityJournal of Science and Technology, in pres.20. Arslan, D. 2018. A Novel Hybrid Method for Singularly Perturbed Delay Differential Equations, Gazi UniversityJournal of Sciences, in pres.21. Boussinesq, J. 1872. Théorie des ondes et de remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, Journal Mathematiques Pures Appliquees, 17, 55-108.22. Duran S., Askin M., Tukur A.S. 2017. New Soliton Properties to the Ill-posed Boussinesq Equation arising in Nonlinear Physical Science, An International Journal of Optimization and Control: Theories & Applications, 7(3): 240-247. 23. Gao, B., Tian, H. 2015. Symmetry Reductions and Exact Solutions to the Ill-Posed Boussinesq, International Journal of Non-Linear Mechanics, 72, 80-83.