Penyelesaian Persamaan Diferensial Abelian Menggunakan Metode Perturbasi Homotopi Transformasi Elzaki
DOI:
https://doi.org/10.26594/jmpm.v6i1.2031Keywords:
Deret polinomial, metode perturbasi homotopi transformasi Elzaki, persamaan diferensial Abelian tipe pertama, simulasi numerik.Abstract
Metode transformasi Elzaki perturbasi homotopi merupakan salah satu metode semi analitik yang dikonstruksi menggunakan kombinasi metode transformasi Elzaki dan metode perturbasi homotopi. Metode ini digunakan untuk menyelesaikan persamaan diferensial, baik linear maupun nonlinear dalam bentuk deret polinomial. Artikel ini membahas penyelesaian persamaan diferensial Abelian jenis pertama menggunakan metode perturbasi homotopi transformasi Elzaki. Simulasi numerik diberikan untuk menguji konvergensi penyelesaian hampiran dengan menggunakan dua contoh. Hasil kajian menunjukkan bahwa metode tersebut dapat menyelesaikan persamaan diferensial Abelian. Nilai-nilai penyelesaian hampiran juga di plot dalam bentuk grafik untuk beberapa jumlah suku-suku solusi hampiran. Hasil plot menunjukkan bahwa pola kurva yang dibentuk cenderung teratur.
References
Adomian, G. (1990). A review of the decomposition method and some recent results for nonlinear equations. Mathematical Computer and Modeling, 13(7), 17–43.
Aldilla, L., Jaharuddin, & Siswandi. (2012). Penggunaan metode perturbasi homotopi untuk menyelesaikan masalah aliran fluida sisko pada pipa lurus. Jurnal Matematika Dan Aplikasinya, 11(1), 11–20.
Bhadane, P. K. G., & Pradhan, V. H. (2013). Elzaki transform homotopy perturbation method for solving gas dynamics equation. International Journal of Research in Engineering and Technology, 2(12), 260–264.
Braun, M. (1992). Differential Equations and Their Applications: An Introduction to Applied Mathematics. Springer. New York.
Elzaki, T. M. (2011). The new integral transform " ELzaki transform ". Global Journal of Pure and Applied Mathematics, 7(1), 57–64.
Elzaki, T. M., & Biazar, J. (2013). Homotopy perturbation method and Elzaki transform for solving system of nonlinear partial differential equations. World Applied Sciences Journal, 24(7), 944–948. https://doi.org/10.5829/ idosi.wasj.2013.24.07.1041
Elzaki, T. M., & Hilal, E. M. A. (2012). Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations. Mathematical Theory and Modeling, 2(3), 33–42.
Ghorbani, A. (2009). Beyond Adomian polynomials : He polynomials. Chaos, Solitons and Fractals, 39, 1486–1492. https://doi.org/10.1016/ j.chaos.2007.06.034
He, J. (1997). A new approach to nonlinear partial differential equations. Communications in Nonlinear Sciences and Numerical Simulation, 2(4), 230–235.
He, J. (1999). Homotopy perturbation technique. Computer Method in Applied Mechanics and Engineering, 178, 257–262.
Hesameddini, E., & Abdollahy, N. (2013). Homotopy perturbation and Elzaki transform for solving Sine-Gorden and Klein-Gorden equations. Iranian Journal of Numerical Analysis and Optimization, 3(2), 33–46.
Jang, M., Chen, C., & Liy, Y.-C. (2000). On solving the initial-value problems using the differential transformation method. Applied Mathematics and Computation, 115, 145–160.
Jaradat, O. K. (2008). Adomian decomposition method for solving Abelian differential equations. Journal of Applied Sciences, 8(10), 1962–1966.
Khatizah, E., Karima, P. T., & Astuti, D. I. (2015). Aplikasi metode transformasi diferensial pada sistem persamaan diferensial biasa. Jurnal Matematika Dan Aplikasinya, 14(2), 1–8.
Liao, S. (2004). On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147, 499–513. https://doi.org/ 10.1016/S0096-3003(02)00790-7
Minarti, N., Kiftiah, M., & Helmi. (2015). Penyelesaian persamaan diferensial parsial linear dengan menggunakan metode transformasi Elzaki. Buletin Ilmiah Matematika Statistika Dan Terapannya, 04(3), 227–236.
Nayfeh, A. H. (1985). Problems in Perturbation. Wiley. New York.
Polyanin, A. D., & Zaitsev, V. F. (1995). Handbook of Solution for Ordinary Differential Equations. CRC Press Inc. Boca Raton.
Schiesser, W. R. (2014). Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equations Applications with R. John Wiley & Sons, Inc. New Jersey.
Singh, P., & Sharma, D. (2020). Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional. Nonlinear Engineering, 9, 60–71. https://doi.org/10.1515/nleng-2018-0136
Suleman, M., Wu, Q., & Abbas, G. (2016). Approximate analytic solution of (2 + 1) dimensional coupled differential Burger’s equation using Elzaki homotopy perturbation method. Alexandria Engineering Journal, 55, 1817–1826. https://doi.org/10.1016/j.aej.2016.03.041
Wartono, Hanafi, M., & Suryani, I. (2020). The solution of nonlinear parabolic equation using variational iteration method. Jurnal Matematika Statistika Dan Komputasi, 16(3), 287–295. https://doi.org/10.20956/jmsk.v16i3.8468
Wartono, & Muda, Y. (2011). Aproksimasi metode dekomposisi adomian pada persamaan diferensial hiperbolik linear. Jurnal Sains, Teknologi Dan Industri, 9(1), 97–103. https://doi.org/10.24014/sitekin.v9i1.612
Wartono, & Muhaijir, M. N. (2013). Penyelesaian persamaan riccati dengan menggunakan metode dekomposisi Adomian Laplace. Jurnal Sains, Teknologi Dan Industri, 10(2), 97–101. https://doi.org/10.24014/ sitekin.v11i1.562
Watugala, G. K. (1993). Sumudu transform : a new integral transform to solve differential equations and control engineering problems. International Journal Mathematics Education Sciences Technology, 24(1), 35–43. https://doi.org/10.1080/0020739930240105
Wazwaz, A., & Mehanna, M. S. (2010). The combined Laplace-Adomian method for handling singular integral equation of heat transfer. International Journal of Nonlinear Science, 10(2), 248–252.
Wibowo, A. T., Jaharuddin, & Kusnanto, A. (2013). Penggunaan metode homotopi untuk menyelesaikan model aliran polutan di tiga danau yang saling terhubung. Jurnal Matematika Dan Aplikasinya, 12(1), 79–92.
Downloads
Published
Issue
Section
License
The formal legal aspect of access to any information and articles contained in this journal website refers to the Creative Commons Attribution 4.0 International (CC BY 4.0) license terms.
