Formulasi Infinitesimal Generators Grup Lie Satu Parameter dari Transformasi Translasi dan Scaling
DOI:
https://doi.org/10.59632/leibniz.v5i02.496Keywords:
Aksi, Grup Lie, Infinitesimal Generators, Scaling, Transformasi, TranslasiAbstract
Grup Lie transformasi dapat dikarakterisasi melalui infinitesimal generators yang membentuk aljabar Lie. Infinitesimal generators dapat diaplikasikan untuk menyelesaikan persamaan diferensial biasa (PDB) maupun persamaan diferensial parsial (PDP) baik yang linear maupun nonlinear. Tujuan penelitian ini adalah untuk memberikan rumus ekplisit infinitesimal generators berkenaan dengan transformasi grup Lie satu parameter. Metode penelitian yang digunakan merupakan kombinasi dari metode kualitiatif berupa studi literatur khususnya transformasi translasi dan scaling dan metode kuantitatif dengan menentukan rumus eksplisit infinitesimal generators dan analisisnya. Hasil yang diperoleh adalah bentuk rumus eksplisit infinitesimal generators yang bersesuaian dengan jenis transformasi yang digunakan. Hasil ini bisa digunakan untuk penelitian selanjutnya dalam menyelesaikan model matematika reaksi difusi konveksi (RDK) dalam PDB maupun PDP sebagai salah satu langkah dalam aplikasi simetri Lie.
Downloads
References
Bildirici, M., Ucan, Y., & Lousada, S. (2022). Interest Rate Based on The Lie Group SO(3) in the Evidence of Chaos. Mathematics, 10(21), 3998. https://doi.org/10.3390/math10213998
Bluman, G. W. , & Cole, J. D. (1969). The general similarity solution of the heat equation. J. Math. Mech., 18(1), 1025–1042.
Bluman, G. W., & Kumei, S. (2002). Symmetry and Integration Methods for Differential Equations (Vol. 154). Springer New York. https://doi.org/10.1007/b97380
Bradshaw-Hajek, B. (2019). Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations. Symmetry, 11(2), 208. https://doi.org/10.3390/sym11020208
Broadbridge, P., & Arrigo, D. J. (1999). All Solutions of Standard Symmetric Linear Partial Differential Equations Have Classical Lie Symmetry. Journal of Mathematical Analysis and Applications, 234(1), 109–122. https://doi.org/10.1006/jmaa.1999.6331
Cheng, Z., Wang, T., Bai, Z., Wang, L., Yuan, C., Zhao, Z., & Song, W. (2023). A numerical simulation investigation on low permeability reservoirs air flooding: Oxidation reaction models and factors. Geoenergy Science and Engineering, 223, 211506. https://doi.org/10.1016/j.geoen.2023.211506
Cherniha, R., & Davydovych, V. (2017). Nonlinear Reaction-Diffusion Systems (Vol. 2196). Springer International Publishing. https://doi.org/10.1007/978-3-319-65467-6
Cherniha, R., Davydovych, V., & King, J. R. (2023). The Shigesada–Kawasaki–Teramoto model: Conditional symmetries, exact solutions and their properties. Communications in Nonlinear Science and Numerical Simulation, 124, 107313. https://doi.org/10.1016/j.cnsns.2023.107313
Cherniha, R., & Pliukhin, O. (2013). New conditional symmetries and exact solutions of reaction–diffusion–convection equations with exponential nonlinearities. Journal of Mathematical Analysis and Applications, 403(1), 23–37. https://doi.org/10.1016/j.jmaa.2013.02.010
Cherniha, R., Serov, M., & Pliukhin, O. (2017). Nonlinear Reaction-Diffusion-Convection Equations. Chapman and Hall/CRC. https://doi.org/10.1201/9781315154848
Cowling, M. G. (2023). Decay estimates for matrix coefficients of unitary representations of semisimple Lie groups. Journal of Functional Analysis, 285(8), 110061. https://doi.org/10.1016/j.jfa.2023.110061
Fisher, R. A. (1937). The Wave of Advance of Advantageous Genes . Annals of Eugenics, 7(4), 355–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
Fisher, R. A. (1999). Natural selection (R. A. Fisher, Ed.). Oxford University PressOxford. https://doi.org/10.1093/oso/9780198504405.001.0001
Freire, I. L., & Torrisi, M. (2014). Weak Equivalence Transformations for a Class of Models in Biomathematics. Abstract and Applied Analysis, 2014, 1–9. https://doi.org/10.1155/2014/546083
Hampsey, M., van Goor, P., Banavar, R., & Mahony, R. (2024). Exploiting Equivariance in the Design of Tracking Controllers for Euler-Poincare Systems on Matrix Lie Groups. IFAC-PapersOnLine, 58(6), 333–338. https://doi.org/10.1016/j.ifacol.2024.08.303
He, W., & Richter, O. (2018). Modelling Large Scale Invasion of Aedes aegypti and Aedes albopictus Mosquitoes. Advances in Pure Mathematics, 08(03), 245–265. https://doi.org/10.4236/apm.2018.83013
Lim, N., & Privault, N. (2016). Analytic bond pricing for short rate dynamics evolving on matrix Lie groups. Quantitative Finance, 16(1), 119–129. https://doi.org/10.1080/14697688.2014.990497
Maidana, N. A., & Yang, H. M. (2008). Describing the geographic spread of dengue disease by traveling waves. Mathematical Biosciences, 215(1), 64–77. https://doi.org/10.1016/j.mbs.2008.05.008
Olver, P. J. (1986). Applications of Lie Groups to Differential Equations (Vol. 107). Springer New York. https://doi.org/10.1007/978-1-4684-0274-2
Orhan, Ö., Torrisi, M., & Tracinà, R. (2019). Group methods applied to a reaction-diffusion system generalizing Proteus Mirabilis models. Communications in Nonlinear Science and Numerical Simulation, 70, 223–233. https://doi.org/10.1016/j.cnsns.2018.10.013
Rosen, G. (1976). The mathematical theory of diffusion and reaction in permeable catalysts: Questions of uniqueness, stability, and transient behavior. Bulletin of Mathematical Biology, 38(1), 95–96. https://doi.org/10.1016/S0092-8240(76)80048-1
Sánchez, R., Grau, R., & Morgado, E. (2006). A novel Lie algebra of the genetic code over the Galois field of four DNA bases. Mathematical Biosciences, 202(1), 156–174. https://doi.org/10.1016/j.mbs.2006.03.017
Torrisi, M., & Traciná, R. (2021). Lie Symmetries and Solutions of Reaction Diffusion Systems Arising in Biomathematics. Symmetry, 13(8), 1530. https://doi.org/10.3390/sym13081530
Zhang, E., & Noakes, L. (2020). Riemannian cubics in quadratic matrix Lie groups. Applied Mathematics and Computation, 375, 125082. https://doi.org/10.1016/j.amc.2020.125082
Zhang, Z.-Y., Zhang, H., Liu, Y., Li, J.-Y., & Liu, C.-B. (2023). Generalized conditional symmetry enhanced physics-informed neural network and application to the forward and inverse problems of nonlinear diffusion equations. Chaos, Solitons & Fractals, 168, 113169. https://doi.org/10.1016/j.chaos.2023.113169
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Edi Kurniadi, Badrulfalah, dan Nurul Gusriani
This work is licensed under a Creative Commons Attribution 4.0 International License.
