najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-1-5-12najo-137Research ArticleMATHEMATICSМАТЕМАТИКАA numerical investigation of modified Burgers’ equation in dusty plasmas with non-thermal ions and trapped electronsЧисленное исследование модифицированного уравнения Бюргерса в пылевой плазме с нетепловыми ионами и захваченными электронамиhttps://orcid.org/0000-0003-4280-3728ДекаХ.DekaH.

Harekrishna Deka,

Khanapara, Guwahati, 781022.

harekrishnadeka11@gmail.comhttps://orcid.org/0000-0002-0793-5680СармаДж.SarmaJ.

Jnanjyoti Sarma,

Fatasil Ambari, Guwahati, 781025.

sjnanjyoti@gmail.comK.K. Handiqui State Open University, Department of MathematicsIndiaR.G. Baruah CollegeIndia202305062025141512Copyright © Deka H., Sarma J., 20252025Дека Х., Сарма Д.Deka H., Sarma J.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/137

In this paper, one-dimensional lower order modified Burgers’ equation (MBE) in dusty plasmas having non-thermal ions and trapped electrons is investigated numerically by finite difference explicit method. The numerical results obtained by the finite difference explicit method for various values of the nonlinear and dissipative coefficients have been compared with the analytical solutions. The obtained numerical results are found to have good agreement with the analytical solutions. It is found that the nonlinear and dissipative coefficients have very important effect on the dust acoustic waves in the system. The absolute error between the analytical and the numerical solutions of the MBE is demonstrated. The stability condition is derived in terms of the equation parameters and the discretization using the von Neumann stability analysis. It has been observed that the waves become flatten and steeper when the dissipative coefficient decreases. It can be concluded that the finite difference explicit method is suitable and efficient method for solving the modified Burgers’ equation.

В данной работе явным методом конечных разностей численно исследуется одномерное модифицированное уравнение Бюргерса (МБЭ) низшего порядка в пылевой плазме, содержащей нетепловые ионы и захваченные электроны. Численные результаты, полученные явным методом конечных разностей при различных значениях нелинейного и диссипативного коэффициентов, сопоставлены с аналитическими решениями. Установлено, что полученные численные результаты хорошо согласуются с аналитическими решениями. Установлено, что очень важное влияние на пылевые акустические волны в системе оказывают нелинейный и диссипативный коэффициенты. Погрешность между аналитическим и численным решениями МЛЭ проявляется в абсолютной погрешности. Условие устойчивости выводится в терминах параметров уравнения и дискретизации с использованием анализа устойчивости по фон Нейману. Было замечено, что волны становятся более плоскими и более крутыми, когда коэффициент рассеяния уменьшается. Можно сделать вывод, что явный метод конечных разностей является подходящим и эффективным методом решения модифицированного уравнения Бюргерса.

плазмапылевая плазманетепловые ионыметод редуктивных возмущениймодифицированное уравнение Бюргерсаявный метод конечных разностейанализ устойчивости фон Неймана.plasmadusty plasmasnon-thermal ionsreductive perturbation methodmodified Burgers’ equationfinite difference explicit methodvon Neumann stability analysisReferencesGoertz C. Dusty plasmas in the solar system. Reviews of Geophysics, 1989, 27 (2), P. 271–292.Goertz C. Dusty plasmas in the solar system. Reviews of Geophysics, 1989, 27 (2), P. 271–292.Adhikary N.C., Deka M.K., Bailung H. Observation of rarefactive ion acoustic solitary waves in dusty plasma containing negative ions. Physics of Plasmas, 2009, 16 (6), 063701.Adhikary N.C., Deka M.K., Bailung H. Observation of rarefactive ion acoustic solitary waves in dusty plasma containing negative ions. Physics of Plasmas, 2009, 16 (6), 063701.Deka M., Adhikary A., Misra A., Bailung H., Nakamura Y. Characteristics of ion-acoustic solitary wave in a laboratory dusty plasma under the influence of ion-beam. Physics of Plasmas, 2012, 19 (10), 103704.Deka M., Adhikary A., Misra A., Bailung H., Nakamura Y. Characteristics of ion-acoustic solitary wave in a laboratory dusty plasma under the influence of ion-beam. Physics of Plasmas, 2012, 19 (10), 103704.Tagare S., Reddy R.V. Effect of higher-order nonlinearity on propagation of nonlinear ion-acoustic waves in a collisionless plasma consisting of negative ions. J. of Plasma Physics, 1986, 35 (2), P. 219–237.Tagare S., Reddy R.V. Effect of higher-order nonlinearity on propagation of nonlinear ion-acoustic waves in a collisionless plasma consisting of negative ions. J. of Plasma Physics, 1986, 35 (2), P. 219–237.El-Labany S. Contribution of higher-order nonlinearity to nonlinear ion-acoustic waves in a weakly relativistic warm plasma. Part 1. Isothermal case. J. of Plasma Physics, 1993, 50 (3), P. 495–504.El-Labany S. Contribution of higher-order nonlinearity to nonlinear ion-acoustic waves in a weakly relativistic warm plasma. Part 1. Isothermal case. J. of Plasma Physics, 1993, 50 (3), P. 495–504.Asgari H., Muniandy S., Wong C. Dust-acoustic shock formation in adiabatic hot dusty plasmas with variable charge. Physics of Plasmas, 2011, 18 (1), 013702.Asgari H., Muniandy S., Wong C. Dust-acoustic shock formation in adiabatic hot dusty plasmas with variable charge. Physics of Plasmas, 2011, 18 (1), 013702.Schamel H. A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons. J. of Plasma Physics, 1973, 9 (3), P. 377–387.Schamel H. A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons. J. of Plasma Physics, 1973, 9 (3), P. 377–387.Mamun A., Cairns R., Shukla P. Effects of vortex-like and nonthermal ion distributions on non-linear dust-acoustic waves. Physics of Plasmas, 1996, 3 (7), P. 2610–2614.Mamun A., Cairns R., Shukla P. Effects of vortex-like and nonthermal ion distributions on non-linear dust-acoustic waves. Physics of Plasmas, 1996, 3 (7), P. 2610–2614.Javidan K., Pakzad H. Obliquely propagating electron acoustic solitons in a magnetized plasma with superthermal electrons. Indian J. of Physics, 2013, 87 (1), P. 83–87.Javidan K., Pakzad H. Obliquely propagating electron acoustic solitons in a magnetized plasma with superthermal electrons. Indian J. of Physics, 2013, 87 (1), P. 83–87.Dev A.N., Deka M.K., Sarma J., Adhikary N.C. Shock wave solution in a hot adiabatic dusty plasma having negative and positive non-thermal ions with trapped electrons. J. of the Korean Physical Society, 2015, 67 (2), P. 339–345.Dev A.N., Deka M.K., Sarma J., Adhikary N.C. Shock wave solution in a hot adiabatic dusty plasma having negative and positive non-thermal ions with trapped electrons. J. of the Korean Physical Society, 2015, 67 (2), P. 339–345.Smith G.D.,Smith G.D., Smith G.D.S. Numerical solution of partial differential equations: finite difference methods. Oxford University Press, 1985.Smith G.D.,Smith G.D., Smith G.D.S. Numerical solution of partial differential equations: finite difference methods. Oxford University Press, 1985.Dehghan M. Parameter determination in a partial differential equation from the overspecified data. Mathematical and Computer Modelling, 2005, 41 (2–3), P. 196–213.Dehghan M. Parameter determination in a partial differential equation from the overspecified data. Mathematical and Computer Modelling, 2005, 41 (2–3), P. 196–213.Bratsos A., Petrakis L. An explicit numerical scheme for the modified burgers’ equation. Int. J. for Numerical Methods in Biomedical Engineering, 2011, 27 (2), P. 232–237.Bratsos A., Petrakis L. An explicit numerical scheme for the modified burgers’ equation. Int. J. for Numerical Methods in Biomedical Engineering, 2011, 27 (2), P. 232–237.Ramadan M.A., El-Danaf T.S. Numerical treatment for the modified burgers equation. Mathematics and Computers in Simulation, 2005, 70 (2), P. 90–98.Ramadan M.A., El-Danaf T.S. Numerical treatment for the modified burgers equation. Mathematics and Computers in Simulation, 2005, 70 (2), P. 90–98.Irk D. Sextic B-spline collocation method for the modified Burgers’ equation. Kybernetes, 2009, 38 (9), P. 1599–1620.Irk D. Sextic B-spline collocation method for the modified Burgers’ equation. Kybernetes, 2009, 38 (9), P. 1599–1620.Aswin V., Awasthi A. Iterative differential quadrature algorithms for modified burgers equation. Engineering Computations, 2018, 35 (1), P. 235– 250.Aswin V., Awasthi A. Iterative differential quadrature algorithms for modified burgers equation. Engineering Computations, 2018, 35 (1), P. 235– 250.Roshan T., Bhamra K. Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method. Applied Mathematics and Computation, 2011, 218 (7), P. 3673–3679.Roshan T., Bhamra K. Numerical solutions of the modified Burgers’ equation by Petrov-Galerkin method. Applied Mathematics and Computation, 2011, 218 (7), P. 3673–3679.Griewank A., El-Danaf T.S. Efficient accurate numerical treatment of the modified Burgers’ equation. Applicable Analysis, 2009, 88 (1), P. 75–87.Griewank A., El-Danaf T.S. Efficient accurate numerical treatment of the modified Burgers’ equation. Applicable Analysis, 2009, 88 (1), P. 75–87.Duan Y., Liu R., Jiang Y. Lattice Boltzmann model for the modified Burgers’ equation. Applied Mathematics and Computation, 2008, 202 (2), P. 489–497.Duan Y., Liu R., Jiang Y. Lattice Boltzmann model for the modified Burgers’ equation. Applied Mathematics and Computation, 2008, 202 (2), P. 489–497.Bashan A., Karakoc S.B.G., Geyikli T. B-spline differential quadrature method for the modified Burgers’ equation. Cankaya University J. of Science and Engineering, 2015, 12 (1), P. 1–13.Bashan A., Karakoc S.B.G., Geyikli T. B-spline differential quadrature method for the modified Burgers’ equation. Cankaya University J. of Science and Engineering, 2015, 12 (1), P. 1–13.Ucar A., Yagmurlu N.M., Tasbozan O. Numerical solutions of the modified Burgers’ equation by finite difference methods. J. of Applied Mathematics, Statistics and Informatics, 2017, 13 (1), P. 19–30.Ucar A., Yagmurlu N.M., Tasbozan O. Numerical solutions of the modified Burgers’ equation by finite difference methods. J. of Applied Mathematics, Statistics and Informatics, 2017, 13 (1), P. 19–30.Karakoc S.B.G., Bashan A., Geyikli T. Two different methods for numerical solution of the modified Burgers’ equation. The Scientific World J., 2014, 2014, 780269.Karakoc S.B.G., Bashan A., Geyikli T. Two different methods for numerical solution of the modified Burgers’ equation. The Scientific World J., 2014, 2014, 780269.Adhikary A., Deka M., Dev A., Sarmah J. Modified Korteweg-de Vries equation in a negative ion rich hot adiabatic dusty plasma with non-thermal ion and trapped electron. Physics of Plasmas, 2014, 21 (8), 083703.[23] equation in a negative ion rich hot adiabatic dusty plasma with non-thermal ion and trapped electron. Physics of Plasmas, 2014, 21 (8), 083703.Schamel H. Stationary solitary, snoidal and sinusoidal ion acoustic waves. Plasma Physics, 1972, 14 (10), 905Schamel H. Stationary solitary, snoidal and sinusoidal ion acoustic waves. Plasma Physics, 1972, 14 (10), 905

The authors declare that there are no conflicts of interest present.