Numerical Simulation and Analysis of the Modified Burgers' Equation in Dusty Plasmas
Abstract
This paper presents a comprehensive study of the numerical simulation of the one-dimensional modified Burgers' equation in dusty plasmas. The reductive perturbation method is employed to derive the equation, and a numerical solution is obtained using the explicit finite difference technique. The obtained results are extensively compared with analytical solutions, demonstrating a high level of agreement, particularly for lower values of the dissipation coefficient. The accuracy and efficiency of the technique are evaluated based on the absolute error. Additionally, the accuracy and effectiveness of the technique are assessed by plotting L2 and L∞ error graphs. The technique's reliability is further confirmed through von Neumann stability analysis, which indicates that the technique is conditionally stable. Overall, the study concludes that the proposed technique is successful and dependable for numerically simulating the modified Burgers' equation in dusty plasmas.
Downloads
References
S. Raut, K.K. Mondal, P. Chatterjee, and A. Roy, ”Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev–Petviashvili–Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution,” SeMA Journal, 78, 571-593 (2021). https://doi.org/10.1007/s40324-021-00242-5
C. Goertz, ”Dusty plasmas in the solar system,” Reviews of Geophysics, 27(2), 271–292 (1989). https://doi.org/10.1029/RG027i002p00271.
D.A. Mendis, and M.Rosenberg, ”Cosmic Dusty Plasma,” Annual Review of Astronomy and Astrophysics, 32(1), 419-463 (1994). https://doi.org/10.1146/annurev.aa.32.090194.002223.
P.K. Shukla, ”A survey of dusty plasma physics,” Physics of Plasmas, 8(5), 1791–1803 (2001). https://doi.org/10.1063/1.1343087.
M. Horanyi, and D.A. Mendis, ”The dynamics of charged dust in the tail of comet Giacobini Zinner,” Journal of Geophysical Research: Space Physics, 91(A1), 355-361 (1986). https://doi.org/10.1029/SP027p0313.
M. Hor´anyi, ”Charged dust dynamics in the solar system,” Annual review of astronomy and astrophysics, 34(1), 383-418 (1996). https://doi.org/10.1146/annurev.astro.34.1.383.
P.K. Shukla, and L. Stenflo, ”Stimulated scattering of electromagnetic waves in dusty plasmas,” Astrophysics and space science, 190(1), 23-32 (1992). https://doi.org/10.1063/1.871450.
J. Tamang, and A. Saha, ”Phase plane analysis of the dust-acoustic waves for the Burgers equation in a strongly coupled dusty plasma,”, Indian Journal of Physics, 95(4), 749-757 (2021). https://DOI:10.1007/s12648-020-01733-3.
A. Barkan, A.N. D’angelo, and R.L. Merlino, ”Experiments on ion-acoustic waves in dusty plasmas,” Planetary and Space Science, 44(1), 239-242 (1996). https://doi.org/10.1016/0032-0633(95)00109-3.
P.K. Shukla, and V.P.Silin, ”Dust ion-acoustic wave,” Physica Scripta, 45(5), 508 (1992). DOI10.1088/0031-8949/45/5/015.
R. Merlino, ”Dusty plasmas: From Saturn’s rings to semiconductor processing devices,” Advances in Physics: X, 6(1), 1873859 (2021). https://doi.org/10.1080/23746149.2021.1873859.
S. Ratynskaia, A. Bortolon, and S.I. Krasheninnikov, ”Dust and powder in fusion plasmas: Recent developments in theory, modeling, and experiments,” Reviews of Modern Plasma Physics, 6(1), 20 (2022). https://doi.org/10.1007/s41614-022-00081-5.
J. Tamang, and A. Saha, ”Influence of dust-neutral collisional frequency and nonextensivity on dynamic motion of dust-acoustic waves,” Waves in Random and Complex Media,” 31(4), 597-617 (2021). https://doi.org/10.1080/17455030.2019.1605230.
A.N. Dev, J. Sarma, and M.K. Deka, ”Dust acoustic shock waves in arbitrarily charged dusty plasma with low and high temperature non-thermal ions,” Canadian Journal of Physics, 93(10), 1030-1038 (2015). https://doi.org/10.1139/cjp-2014-0391.
R. Tian, L. Fu, Y. Ren, and H. Yang, ”(3+1)-Dimensional time-fractional modified Burgers equation for dust ionacoustic waves as well as its exact and numerical solutions,” Mathematical Methods in the Applied Sciences, 44(10), 8177-8196 (2021). https://doi.org/10.1002/mma.5823.
U. Yusuf, M. Ya˘gmurlu, and A. Bashan, ”Numerical solutions and stability analysis of modified Burgers equation via modified cubic B-spline differential quadrature methods,” Sigma Journal of Engineering and Natural Sciences, 37(1), 129-142 (2019). https://sigma.yildiz.edu.tr/storage/upload/pdfs/1635837147-en.pdf
O. Oru¸c, Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation. International Journal of Computer Mathematics, 98(3), 461-479 (2021). https://doi.org/10.1080/00207160.2020.1755432.
A. Zeytinoglu, M. Sari, and B. Allahverdiev, ”Numerical simulations of shock wave propagating by a hybrid approximation based on high-order finite difference schemes,” Acta Physica Polonica A, 133(1), 140-151 (2018). https://doi.org/10.12693/aphyspola.133.140.
A.G. Bratsos, ”A fourth-order numerical scheme for solving the modified Burgers’ equation,” Computers and Mathematics with Applications, 60(5), 1393-1400 (2010). https://doi.org/10.1016/j.camwa.2010.06.021.
M.A. Ramadan, T.S. El-Danaf, and F.E. Abd Alaal, ”A numerical solution of the Burgers’ equation using septic Bsplines,” Chaos, Solitons and Fractals, 26(4), 1249-1258 (2005). https://doi.org/10.1016/j.chaos.2005.01.054.
D. Irk, ”Sextic B spline collocation method for the modified Burgers’ equation,” Kybernetes, 38(9), 1599-1620 (2009). https://doi.org/10.1108/03684920910991568.
B. Saka, and I. Dag, ”A numerical study of the Burgers’ equation,” Journal of the Franklin Institute, 345(4), 328-348 (2008). https://doi.org/10.1016/j.jfranklin.2007.10.004.
Y. Duan, R. Liu, and Y. Jiang, ”Lattice Boltzmann model for the modified Burgers’ equation,” Applied Mathematics and Computation, 202(2), 489-497 (2008). https://doi.org/10.1016/j.amc.2008.01.020
R.S. Temsah, ”Numerical solutions for convection-diffusion equation using El-Gendi method,” Communications in Nonlinear Science and Numerical Simulation, 14(3), 760-769 (2009). https://doi.org/10.1016/j.cnsns.2007.11.004
T. Roshan, and K.S. Bhamra, ”Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method,” Applied Mathematics and Computation, 218(7), 3673-3679 (2011). https://doi.org/10.1016/j.amc.2011.09.010
S. Kutluay, Y. Ucar, and N.M. Yagmurlu, ”Numerical solutions of the modified Burgers’ equation by a cubic B-spline collocation method,” Bulletin of the Malaysian Mathematical Sciences Society, 39(4), 1603-1614 (2016). https://doi.org/10.1016/j.amc.2012.01.059
Y. Ucar, N.M. Yagmurlu, and O. Tasbozan, ”Numerical solutions of the modified Burgers’ equation by finite difference methods,” Journal of applied mathematics, statistics and informatics, 13(1), 19-30 (2017). https://doi.org/10.1515/jamsi-2017-0002
W. Gao, Y. Liu, B. Cao, and H. Li, ”A High-Order NVD/TVD-Based Polynomial Upwind Scheme for the Modified Burgers’ Equations,” Advances in Applied Mathematics and Mechanics, 4(5), 617-635 (2012). https://doi.org/10.4208/aamm.10-m1139
A. Griewank, and T.S. El-Danaf, ”Efficient accurate numerical treatment of the modified Burgers’ equation,” Applicable Analysis, 88(1), 75-87 (2009). https://doi.org/10.1080/00036810802556787
A.G. Bratsos, and L.A. Petrakis, ”An explicit numerical scheme for the modified Burgers’ equation,” International Journal for Numerical Methods in Biomedical Engineering, 27(2), 232-237 (2011). https://doi.org/10.1002/cnm.1294
Shallu, and V.K. Kukreja, ”An improvised collocation algorithm with specific end conditions for solving modified Burgers equation,” Numerical Methods for Partial Differential Equations, 37(1), 874-896 (2021). https://doi.org/10.1002/num.22557
A. Kumari, and V.K. Kukreja, ”Error bounds for septic Hermite interpolation and its implementation to study modified Burgers’ equation,” Numerical Algorithms, 89(4), 1799-1821 (2022). https://doi.org/10.1007/s11075-021-01173-y
L. Chandrasekharan Nair, and A. Awasthi, ”Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers’ equation,” Numerical Methods for Partial Differential Equations, 35(3), 1269-1289 (2019). https://doi.org/10.1002/num.22349
G.W. Recktenwald, ”Finite-difference approximations to the heat equation,” Mechanical Engineering, 10 (01) (2004). https://webspace.science.uu.nl/~zegel101/MOLMODWISK/FDheat2.pdf
M.R. Jana, A. Sen, and P.K. Kaw, ”Collective effects due to charge-fluctuation dynamics in a dusty plasma,” Physical Review E, 48(5), 3930 (1993). https://doi.org/10.1103/PhysRevE.48.3930
J.R. Bhatt, and B.P. Pandey, ”Self-consistent charge dynamics and collective modes in a dusty plasma,” Phys. Rev. E, 50(5), 3980–3983 (1994). https://doi.org/10.1103/PhysRevE.50.3980
N. Parumasur, R.A. Adetona, and P. Singh, ”Efficient solution of burgers’, modified burgers’ and KdV–burgers’,”Mathematics, 11(8), 1847 (2023). https://doi.org/10.3390/math11081847
Copyright (c) 2023 Harekrishna Deka, Jnanjyoti Sarma
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
