Approximate solutions of the fractional Harry Dym equation

Yıl 2024, Cilt: 42 Sayı: 5, 1604 - 1611, 04.10.2024

Sevil Çulha Ünal

Öz

In this paper, the approximate solutions of the time fractional Harry Dym equation with fractional derivative in the Caputo sense are obtained by using the Residual power series method (RPSM). This equation is a significant dynamical equation that occurs in a variety of physical systems. The suggested method provides good accuracy for the approximate solution when compared numerically with the exact solution. The effectiveness of the proposed method is also illustrated with the aid of numerical results. These results indicate that the RPSM is a power, useful, and applicable for determining the solutions of the time Hary Dym equation. Some of these results are illustrated by 2D and 3D graphics. Besides, the proposed method can be applied to many different differential equations due to its ease of use and reliability.

Anahtar Kelimeler

Approximate Solution, Fractional Partial Differential Equation, Caputo Derivative, Harry Dym Equation, Residual Power Series Method

Kaynakça

• REFERENCES
• [1] Vasconcelos GL, Kadanoff LP. Stationary solutions for the Saffman-Taylor problem with surface tension. Phys Rev A 1991;44:16. [CrossRef]
• [2] Kelleche A, Tatar N-E. Control and exponential stabilization for the equation of an axially moving viscoelastic strip. Math Methods Appl Sci 2017;40:62396253. [CrossRef]
• [3] Kelleche A. Boundary control and stabilization of an axially moving viscoelastic string under a boundary disturbance. Math Model Anal 2017;22:763784. [CrossRef]
• [4] Kelleche A, Tatar N-E. Existence and stabilization of a Kirchhoff moving string with a distributed delay in the boundary feedback. Math Model Nat Phenom 2017;12:106117. [CrossRef]
• [5] Kelleche A, Tatar N-E. Adaptive boundary stabilization of a nonlinear axially moving string. Z Angew Math Mech 2021;101:e202000227. [CrossRef]
• [6] Hereman W, Banerjee PP, Chatterjee MR. On the nonlocal equations and nonlocal charges associated with the Harry-Dym hierarchy Korteweg-de Vries equation. J Phys A Math Theor 1989;22:241252. [CrossRef]
• [7] Haghighatdoost G, Bazghandi M. Differential invariants of Harry-Dym equation. The 11th Seminar on Geometry and Topology Yasouj University. 2021;14.
• [8] Mokhtari R. Exact solutions of the Harry–Dym equation. Commun Theor Phys 2011;55:204208. [CrossRef]
• [9] Tian K, Cui M, Liu QP. A note on Bäcklund transformations for the Harry Dym equation. Partial Differ Equ Appl Math 2022;5:100352. [CrossRef]
• [10] Gonzàlez-Gaxiola O, Ruiz de Chàvez J, Edeki SO. Iterative method for constructing analytical solutions to the Harry-Dym initial value problem. Int J Appl Math 2018;31:627640. [CrossRef]
• [11] Singh I, Kumar S. Haar wavelet methods for numerical solutions of Harry Dym (HD), BBM Burger’s and 2D diffusion equations. Bull Braz Math Soc 2018;49:313338. [CrossRef]
• [12] Soltani D, Khorshidi MA. Application of homotopy perturbation and reconstruction of variational iteration methods for Harry Dym equation and compared with exact solution. Int J Multidiscip Curr Res 2013;166169.
• [13] Halim AA. Soliton solutions of the (2+1)-dimensional Harry Dym equation via Darboux transformation. Chaos Solit Fract 2008;36:646653. [CrossRef]
• [14] Xiao Y, Fan E. Long time behavior and soliton solution for the Harry Dym equation. J Math Anal Appl 2019;480:123248. [CrossRef]
• [15] Ahmad B, Nieto JJ. Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations. Abstr Appl Anal 2009;2009:494720. [CrossRef]
• [16] Wang Y, Liang S, Wang Q. Existence results for fractional differential equations with integral and multi-point boundary conditions. Boundary Value Probl 2018;2018:4. [CrossRef]
• [17] Şenol M, Ata A. Approximate solution of time-fractional KdV equations by residual power series method. J Balikesir Univ Sci Technol 2018;20:430439. [CrossRef]
• [18] Akram G, Sadaf M, Abbas M, Zainab I, Gillani SR. Efficient techniques for traveling wave solutions of time-fractional Zakharov-Kuznetsov equation. Math Comput Simul 2022;193:607622. [CrossRef]
• [19] Qurashi MMA, Korpinar Z, Baleanu D, Inc M. A new iterative algorithm on the time-fractional Fisher equation: Residual power series method. Adv Mech Eng 2017;9:18. [CrossRef]
• [20] Körpınar Z. The residual power series method for solving fractional Klein-Gordon equation. Sakarya Univ J Sci 2017;21:285293. [CrossRef]
• [21] Al-Khaled K, Alquran M. An approximate solution for a fractional model of generalized Harry Dym equation. Math Sci 2014;8:125130. [CrossRef]
• [22] Kumar D, Singh J, Kılıçman A. An efficient approach for fractional Harry Dym equation by using Sumudu transform. Abstr Appl Anal 2013;2013:608943. [CrossRef]
• [23] Alshammari S, Iqbal N, Yar M. Analytical investigation of nonlinear fractional Harry Dym and Rosenau-Hyman equation via a novel transform. J Funct Spaces 2022;2022:8736030. [CrossRef]
• [24] Assabaai MA, Mukherij OF. Exact solutions of the Harry Dym equation using Lie group method. Univ Aden J Nat Appl Sci 2020;24:487493. [CrossRef]
• [25] Costa FS, Soares JCA, Plata ARG, Oliveira EC. On the fractional Harry Dym equation. Comput Appl Math 2018;37:28622876. [CrossRef]
• [26] Yue C, Liu G, Li K, Dong H. Similarity solutions to nonlinear diffusion/Harry Dym fractional equations. Adv Math Phys 2021;2021:6670533. [CrossRef]
• [27] Ghiasi EK, Saleh R. A mathematical approach based on the homotopy analysis method: Application to solve the nonlinear Harry-Dym (HD) equation. Appl Math 2017;8:15461562. [CrossRef]
• [28] Shunmugarajan B. An efficient approach for fractional Harry Dym equation by using homotopy analysis method. Int J Eng Res Technol 2016;5:561566. [CrossRef]
• [29] Huang Q, Zhdanov R. Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann-Liouville derivative. Physica A 2014;409:110118. [CrossRef]
• [30] Kumar S, Tripathi MP, Singh OP. A fractional model of Harry Dym equation and its approximate solution. Ain Shams Eng J 2013;4:111115. [CrossRef]
• [31] Nadeem M, Li Z, Alsayyad Y. Analytical approach for the approximate solution of Harry Dym equation with Caputo fractional derivative. Math Probl Eng 2022;2022:4360735. [CrossRef]
• [32] Rawashdeh MS. A new approach to solve the fractional Harry Dym equation using the FRDTM. Int J Pure Appl Math 2014;95:553566. [CrossRef]
• [33] Yokuş A, Gülbahar S. Numerical solutions with linearization techniques of the fractional Harry Dym equation. Appl Math Nonlinear Sci 2019;4:3542. [CrossRef] [34] Iyiola OS, Gaba YU. An analytical approach to time-fractional Harry Dym equation. Appl Math Inf Sci 2016;10:409412. [CrossRef]
• [35] Wang L, Wang D, Shen S, Huang Q. Lie point symmetry analysis of the Harry-Dym type equation with Riemann-Liouville fractional derivative. Acta Math Appl Sin Engl Ser 2018;34:469477. [CrossRef]
• [36] Podlubny I. Fractional Differential Equations. New York: Academic Press; 1999.
• [37] El-Ajou A, Arqub OA, Zhour ZA, Momani S. New results on fractional power series: theories and applications. Entropy 2013;15:53055323.
• [38] El-Ajou A, Arqub OA, Momani S. Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm. J Comput Phys 2015;293:8195. [CrossRef]
• [39] Arqub A. Series solution of Fuzzy differential equations under strongly generalized differentiability. J Adv Res Appl Math 2013;5:3152. [CrossRef]
• [40] Arqub OA, Abo-Hammour Z, Al-Badarneh R, Momani S. A reliable analytical method for solving higher-order initial value problems. Discrete Dyn Nat Soc 2013;2013:673829. [CrossRef]
• [41] Arqub OA, El-Ajou A, Zhour ZA, Momani S. Multiple solutions of nonlinear boundary value problems of fractional order: A new analytic iterative technique. Entropy 2014;16:471493. [CrossRef]
• [42] Arqub OA, El-Ajou A, Bataineh AS, Hashim I. A representation of the exact solution of generalized Lane-Emden equations using a new analytical method. Abstr Appl Anal 2013;2013:378593. [CrossRef]