TY  - JOUR
T1  - Approximate Solutions of Singularly Perturbed Nonlinear Ill-posed and Sixth-order Boussinesq Equations with Hybrid Method
TT  - Hibrit Metot ile Singüler Pertürbe Nonlineer Ill-posed ve Altıncı Mertebe Boussinesq Denklemlerinin Yaklaşık Çözümleri
AU  - Arslan, Derya
PY  - 2019
DA  - June
Y2  - 2019
DO  - 10.17798/bitlisfen.491847
JF  - Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
PB  - Bitlis Eren University
WT  - DergiPark
SN  - 2147-3129
SP  - 451
EP  - 458
VL  - 8
IS  - 2
LA  - en
AB  - The aim of this paper is to obtain the approximate solution of singularly perturbed ill-posed and sixth-orderBoussinesq equation by hybrid method (differential transform and finite difference method) as a differentalternative 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 theefficiency and reliability of the hybrid method. Numerical results are given and compared with exact solution andin literature RDTM solution. The numerical data show that hybrid method is a powerful, quite efficient and ispractically well suited for solving nonlinear singular perturbed Boussinesq equations.
KW  - Sixth-order Boussinesq Equation
KW  - Differential Transform Method
KW  - Finite Difference Method
KW  - Approximate Solution
N2  - Bu çalışmanın amacı, singüler pertürbe lineer olmayan ill-posed ve altıncı mertebeden Boussinesq denklemininfarklı bir alternatif yöntem olan hibrit metotla (diferansiyel dönüşüm ve sonlu fark metodu) yaklaşık çözümünüelde etmektir. 𝑡 −zaman değişkeni için diferansiyel dönüşüm metodu ve 𝑥 −konum değişkeni için sonlu farkmetodu (merkezi fark yaklaşımı) uygulanmıştır. Hibrit yöntemin etkinliğini ve güvenilirliğini göstermek için ikiörnek 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 Boussinesqdenklemlerini çözmek için pratik olarak uygun olduğunu göstermektedir.
CR  - 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.
UR  - https://doi.org/10.17798/bitlisfen.491847
L1  - https://dergipark.org.tr/en/download/article-file/746920
ER  -