Composite Laguerre pseudospectral method for Fokker-Planck equations
Abstract
A composite generalized Laguerre pseudospectral method for the nonlinear Fokker-Planck equations on the whole line is developed. Some composite generalized Laguerre interpolation approximation results are established. As an application, a composite Laguerre pseudospectral scheme is provided for the problems of the relaxation of fermion and boson gases. Convergence and stability of the scheme are proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis.
Keyword : composite generalized Laguerre pseudospectral method, nonlinear FokkerPlanck equations, the whole line
How to Cite
Wang, C., Wang, T., & Shang, Y. (2023). Composite Laguerre pseudospectral method for Fokker-Planck equations. Mathematical Modelling and Analysis, 28(4), 542–560. https://doi.org/10.3846/mma.2023.17513
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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T. Sun and T.-J. Wang. Multi-domain decomposition pseudospectral method for nonlinear Fokker-Planck equations. Communications on Applied Mathematics and Computation, 1(2):231–252, 2019. https://doi.org/10.1007/s42967-019-00013-0
T.-J. Wang. Composite generalized Laguerre spectral method for nonlinear Fokker-Planck equation on the whole line. Mathematical Methods in the Applied Sciences, 40(5):1462–1474, 2017. https://doi.org/10.1002/mma.4067
T.-J. Wang and G. Chai. A fully discrete pseudospectral method for the nonlinear Fokker-Planck equations on the whole line. Applied Numerical Mathematics, 174:17–33, 2022. https://doi.org/10.1016/j.apnum.2022.01.003
T.-J. Wang and B.-Y. Guo. Composite generalized Laguerre-Legendre pseudospectral method for Fokker-Planck equation in an infinite channel. Applied Numerical Mathematics, 58(10):1448–1466, 2008. https://doi.org/10.1016/j.apnum.2007.08.007
Z.-Q. Wang and B.-Y. Guo. Jacobi rational approximation and spectral method for differential equations of degenerate type. Mathematics of Computation, 77(262):883–907, 2008. https://doi.org/10.1090/S0025-5718-07-02074-1
Z.-Q. Wang and C.-T. Sheng. An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Mathematics of Computation, 85(298):635–666, 2016. https://doi.org/10.1090/mcom/3023
Z.Q. Wang, Y.L. Guo and L.J. Yi. An hp-version Legendre-Jacobi spectral collocation method for Volterra integral-differential equations with smooth and weak singular kernels. Mathematics of Computation, 86(307):2285–2324, 2017. https://doi.org/10.1090/mcom/3183
X.-M. Xiang and Z.-Q. Wang. Generalized Hermite spectral method and its applications to problems in unbounded domains. SIAM Journal on Numerical Analysis, 48(4):1231–1253, 2010. https://doi.org/10.1137/090773581
L.J. Yi and B.Q. Guo. An h-p version continuous Petrov-Galerkin finite element method for Volterra integral-differential equations with smooth and nonsmooth singular kernels. SIAM Journal on Numerical Analysis, 53(6):2677–2704, 2015. https://doi.org/10.1137/15M1006489
J. Biazar, P. Gholamin and K. Hosseini. Variational iteration method for solving Fokker-Planck equation. Journal of the Franklin Institute, 347(7):1137–1147, 2010. https://doi.org/10.1016/j.jfranklin.2010.04.007
J.A. Carrillo, J. Rosado and F. Salvarani. 1d nonlinear Fokker-Planck equations for fermions and bosons. Applied Mathematics Letters, 21(2):148–154, 2008. https://doi.org/10.1016/j.aml.2006.06.023
G. Chai and T.J. Wang. Generalized Hermite spectral method for nonlinear Fokker-Planck equations on the whole line. Journal of Mathematical Study, 51(2):177–195, 2018. https://doi.org/10.4208/jms.v51n2.18.04
M. Escobedo and S. Mischler. On a quantum Boltzmann equation for a gas of photons. Journal de Math´ematiques Pures et Appliqu´ees, 80(5):471–515, 2001. https://doi.org/10.1016/S0021-7824(00)01201-0
J.C.M. Fox, B.Y. Guo and T. Tang. Combined Hermite spectral-finite difference method for the Fokker-Planck equation. Mathematics of Computation, 71(240):1497–1528, 2001. https://doi.org/10.1090/S0025-5718-01-01365-5
T.D. Frank. Nonlinear Fokker-Planck equations: Fundanmentals and applications. Springer Series in Synergetics, Springer-Verlag, Berlin, 2005. https://doi.org/10.1007/b137680
B.-Y. Guo, L.-L. Wang and Z.-Q. Wang. Generalized Laguerre interpolation and pseudospectral method for unbounded domains. SIAM Journal on Numerical Analysis, 43(6):2567–2589, 2006. https://doi.org/10.1137/04061324X
B.-Y. Guo and T.-J. Wang. Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel. Mathematics of Computation, 78(265):129–151, 2009. https://doi.org/10.1090/S0025-5718-08-02152-2
B.Y. Guo. Spectral Methods and Their Applications. World Scientific, Singapore, 1998. https://doi.org/10.1142/3662
B.Y. Guo. Error estimation of Hermite spectral method for nonlinear partial differential equations. Mathematics of Computation, 68(227):1067–1078, 1999. https://doi.org/10.1090/S0025-5718-99-01059-5
B.Y. Guo. Spectral and pseudospectral methods for unbounded domains. SCIENTIA SINICA Mathematica, 45(7):975–1024, 2015. https://doi.org/10.1360/N012014-00149
H.J. Hwang, J. Jang and J. Jung. The Fokker-Planck equation with absorbing boundary conditions in bounded domains. SIAM Journal on Mathematical Analysis, 50(2):2194–2232, 2018. https://doi.org/10.1137/16M1109928
G. Kaniadakis. Generalized Boltzmann equation describing the dynamics of bosons and fermions. Physics Letters A, 203(4):229–234, 1995. https://doi.org/10.1016/0375-9601(95)00414-X
X.G. Lu. On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles. Journal of Statistical Physics, 105(1):353–388, 2001. https://doi.org/10.1023/A:1012282516668
S. Martinez, A.R. Plastino and A. Plastino. Nonlinear Fokker-Planck equations and generalized entropies. Physica A: Statistical Mechanics and its Applications, 259(1):183–192, 1998. https://doi.org/10.1016/S0378-4371(98)00277-5
G. Mastroianni and D. Occorsio. Lagrange interpolation at Laguerre zeros in some weighted uniform spaces. Acta Mathematica Hungarica, 91(1):27–52, 2001. https://doi.org/10.1023/A:1010678709857
Ch.-T. Sheng, Z.-Q. Wang and B.-Y. Guo. A multistep LegendreGauss spectral collocation method for nonlinear Volterra integral equations. SIAM Journal on Numerical Analysis, 52(4):1953–1980, 2014. https://doi.org/10.1137/130915200
T. Sun and T.-J. Wang. Multi-domain decomposition pseudospectral method for nonlinear Fokker-Planck equations. Communications on Applied Mathematics and Computation, 1(2):231–252, 2019. https://doi.org/10.1007/s42967-019-00013-0
T.-J. Wang. Composite generalized Laguerre spectral method for nonlinear Fokker-Planck equation on the whole line. Mathematical Methods in the Applied Sciences, 40(5):1462–1474, 2017. https://doi.org/10.1002/mma.4067
T.-J. Wang and G. Chai. A fully discrete pseudospectral method for the nonlinear Fokker-Planck equations on the whole line. Applied Numerical Mathematics, 174:17–33, 2022. https://doi.org/10.1016/j.apnum.2022.01.003
T.-J. Wang and B.-Y. Guo. Composite generalized Laguerre-Legendre pseudospectral method for Fokker-Planck equation in an infinite channel. Applied Numerical Mathematics, 58(10):1448–1466, 2008. https://doi.org/10.1016/j.apnum.2007.08.007
Z.-Q. Wang and B.-Y. Guo. Jacobi rational approximation and spectral method for differential equations of degenerate type. Mathematics of Computation, 77(262):883–907, 2008. https://doi.org/10.1090/S0025-5718-07-02074-1
Z.-Q. Wang and C.-T. Sheng. An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays. Mathematics of Computation, 85(298):635–666, 2016. https://doi.org/10.1090/mcom/3023
Z.Q. Wang, Y.L. Guo and L.J. Yi. An hp-version Legendre-Jacobi spectral collocation method for Volterra integral-differential equations with smooth and weak singular kernels. Mathematics of Computation, 86(307):2285–2324, 2017. https://doi.org/10.1090/mcom/3183
X.-M. Xiang and Z.-Q. Wang. Generalized Hermite spectral method and its applications to problems in unbounded domains. SIAM Journal on Numerical Analysis, 48(4):1231–1253, 2010. https://doi.org/10.1137/090773581
L.J. Yi and B.Q. Guo. An h-p version continuous Petrov-Galerkin finite element method for Volterra integral-differential equations with smooth and nonsmooth singular kernels. SIAM Journal on Numerical Analysis, 53(6):2677–2704, 2015. https://doi.org/10.1137/15M1006489