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

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. ? −zaman değişkeni için diferansiyel dönüşüm metodu ve ? −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.

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

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 ? −time variable and the finite difference method(central difference approach) is applied for ? −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.

PDF

___

[1] 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.
[2] 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.
[3] Dash R.K., Daripa P. 2002. Analytical and Numerical Studies of Singularly Perturbed Boussinesq Equation, Applied Mathematics and Computation, 126: 1–30.
[4] Feng Z. 2003. Traveling Solitary Wave Solutions to the Generalized Boussinesq Equation. Wave Motion, 37 (1): 17–23.
[5] 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.
[6] 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.
[7] Yilmazer R., Bas E. 2012. Explicit Solutions of Fractional Schrödinger Equation via Fractional Calculus Operators, International Journal of Open Problems in Computer Science and Mathematics, 238: 1-10.
[8] Ayaz F. 2004. Solution of the System of Differential Equations by Differential Transform Method. Appl. Math. Comput., 147: 547-567.
[9] Ayaz F. 2004. Applications of Differential Transform Method to Differential Algebraic Equations. Appl. Math. Comput., 152: 649-657.
[10] 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.
[11] Gao B., Tian H. 2015. Symmetry Reductions and Exact Solutions to the Ill-Posed Boussinesq. International Journal of Non-Linear Mechanics, 72: 80-83.
[12] Süngü İ., Demir H. 2012. Solutions of the System of Differential Equations by Differential Transform / Finite Difference Method. Nwsa-Physical Sciences, 7: 1308-7304.
[13] Süngü İ., Demir H. 2012. Application of the Hybrid Differential Transform Method to the Nonlinear Equations. Applied Mathematics, 3: 246-250.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] Arslan D. 2019. A Novel Hybrid Method for Singularly Perturbed Delay Differential Equations. Gazi University Journal of Sciences, 32 (1): 217-223.
[19] Bas E., Metin Türk, F. 2018. An Application of Comparison Criteria to Fractional Spectral Problem Having Coulomb Potential. ThermalScience, 22 (1): 79-S85.
[20] Bas E., Ozarslan R. 2018. Real World Applications of Fractional Models by Atangana–Baleanu Fractional Derivative. Chaos, Solitons & Fractals, 116: 121-125
[21] Bas E., Ozarslan R., Baleanu D., Ercan A. 2018. Comparative Simulations for Solutions of Fractional Sturm–Liouville Problems with Nonsingular Operators, Advances in Difference Equations, 2018 (1): 350.
[22] Bas E., Acay B., Ozarslan R. 2019. Fractional Models with Singular and Non-Singular Kernels for Energy Efficient Buildings. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2): 023110.
[23] Bas E., Ozarslan R. 2017. Sturm-Liouville Problem via Coulomb Type in Difference Equations. Filomat, 31 (4): 989-998.
[24] 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.
[25] Arslan D. 2018. Differential Transform Method for Singularly Perturbed Singular Differential Equations. Journal of the Institute of Natural &Applied Sciences, 23 (3): 254-260.
[26] Zhou J.K. 1986. Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press, Wuhan, China.
[27] 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.
[28] Arslan D. 2018. The Approximate Solution of Fokker-Planck Equation with Reduced Differential Transform Method. Erzincan UniversityJournal of Science and Technology, 12 (1): 281-293.
[29] Mohyud-Din S.T., Muhammad Aslam N. 2009. Homotopy Perturbation Method for Solving Partial Differential Equations. Zeitschrift Naturforschung, 64: 157-170.