September  2017, 10(3): 669-688. doi: 10.3934/krm.2017027

Finite range method of approximation for balance laws in measure spaces

Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: Piotr Gwiazda

Received  April 2016 Revised  July 2016 Published  December 2016

In the following paper we reconsider a numerical scheme recently introduced in [10]. The method was designed for a wide class of size structured population models with a nonlocal term describing the birth process. Despite its numerous advantages it features the exponential growth in time of the number of particles constituting the numerical solution. We introduce a new algorithm free from this inconvenience. The improvement is based on the application the Finite Range Approximation to the nonlocal term. We prove the convergence of the derived method and provide the rate of its convergence. Moreover, the results are illustrated by numerical simulations applied to various test cases.

Citation: Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
References:
[1]

L. M. AbiaO. Angulo and J. C. Lopez-Marcos, Numerical schemes for a size-structured cell population model with equal fission, Mathematical and Computer Modelling, 50 (2009), 653-664.  doi: 10.1016/j.mcm.2009.05.023.  Google Scholar

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, Journal of Differential Equations, 217 (2005), 431-455.  doi: 10.1016/j.jde.2004.12.013.  Google Scholar

[3]

A. S. AcklehB. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 917-928.  doi: 10.3934/dcdsb.2005.5.917.  Google Scholar

[4]

A. L. BertozziT. KolokonikovH. Sun and D. Uminsky, Stability of ring patterns arising from 2d particle interactions, Physical Review E, 84 (2011).   Google Scholar

[5]

C. K. Birdsal and A. B. Langdon, Plasma Physics Via Computer Simulation, McGraw-Hill, New York, 1985. Google Scholar

[6]

A. BrannstromL. Carlsson and D. Simpson, On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013), 3213-3231.  doi: 10.1137/120893215.  Google Scholar

[7]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Leture Series in Mathematics and its Applications vol. 20, Oxford University Press, 2000.  Google Scholar

[8]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[9]

J. A. CarrilloR. ColomboP. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equations, 252 (2012), 3245-3277.  doi: 10.1016/j.jde.2011.11.003.  Google Scholar

[10]

J.A. CarrilloP. Gwiazda and A. Ulikowska, Splitting particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014), 2171-2197.  doi: 10.1142/S0218202514500183.  Google Scholar

[11]

R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753.  doi: 10.3934/dcds.2009.23.733.  Google Scholar

[12]

G. H. Cottet and P. A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal., 21 (1984), 52-76.  doi: 10.1137/0721003.  Google Scholar

[13]

A. M. de Roos, Numerical methods for structured population models: The escalator boxcar train, Numerical Methods for Partial Differential Equations, 4 (1988), 173-195.  doi: 10.1002/num.1690040303.  Google Scholar

[14]

A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Monographs in Population Biology 51, Princeton University Press, Princeton, 2013. Google Scholar

[15]

O. Diekmann and Ph. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Differential Equations, 215 (2005), 268-319.  doi: 10.1016/j.jde.2004.10.025.  Google Scholar

[16]

M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006). Google Scholar

[17]

J. EversS. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097.  doi: 10.1016/j.jde.2015.02.037.  Google Scholar

[18]

K. Ganguly and H. D. Victory Jr, On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989), 249-288.  doi: 10.1137/0726015.  Google Scholar

[19]

J. GoodmanT. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the 2-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430.  doi: 10.1002/cpa.3160430305.  Google Scholar

[20]

P. GwiazdaJ. JabƚońskiA. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014), 1797-1820.  doi: 10.1002/num.21879.  Google Scholar

[21]

P. GwiazdaG. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM Journal on Mathematical Analysis, 44 (2012), 1103-1133.  doi: 10.1137/11083294X.  Google Scholar

[22]

P. GwiazdaT. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.  Google Scholar

[23]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773.  doi: 10.1142/S021989161000227X.  Google Scholar

[24]

F. H. Harlow, The particle-in-cell computing method for fluid dynamics, Methods in computational physics, 3 (1964), 319-343.   Google Scholar

[25]

D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1996), 2099-2119.  doi: 10.1137/S0036142994266856.  Google Scholar

[26]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x.  Google Scholar

[27]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[28]

P. A. Raviart, An analysis of particle methods, Numerical Methods in Fluid Dynamics, Lecture Notes in Math. , Springer, Berlin, 1127 (1985), 243-324. doi: 10.1007/BFb0074532.  Google Scholar

[29]

E. Tadmor, A review of numerical methods for nonlinear partial differential equations, Bulletin of the American Mathematical Society, 49 (2012), 507-554.  doi: 10.1090/S0273-0979-2012-01379-4.  Google Scholar

[30]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate studies in mathematics, American Mathematical Society, 2003. doi: 10.1007/b12016.  Google Scholar

[31]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985.  Google Scholar

[32]

M. Westdickenberg and J. Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations, M2AN Math. Model. Numer. Anal., 44 (2010), 133-166.  doi: 10.1051/m2an/2009043.  Google Scholar

show all references

References:
[1]

L. M. AbiaO. Angulo and J. C. Lopez-Marcos, Numerical schemes for a size-structured cell population model with equal fission, Mathematical and Computer Modelling, 50 (2009), 653-664.  doi: 10.1016/j.mcm.2009.05.023.  Google Scholar

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, Journal of Differential Equations, 217 (2005), 431-455.  doi: 10.1016/j.jde.2004.12.013.  Google Scholar

[3]

A. S. AcklehB. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 917-928.  doi: 10.3934/dcdsb.2005.5.917.  Google Scholar

[4]

A. L. BertozziT. KolokonikovH. Sun and D. Uminsky, Stability of ring patterns arising from 2d particle interactions, Physical Review E, 84 (2011).   Google Scholar

[5]

C. K. Birdsal and A. B. Langdon, Plasma Physics Via Computer Simulation, McGraw-Hill, New York, 1985. Google Scholar

[6]

A. BrannstromL. Carlsson and D. Simpson, On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013), 3213-3231.  doi: 10.1137/120893215.  Google Scholar

[7]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Leture Series in Mathematics and its Applications vol. 20, Oxford University Press, 2000.  Google Scholar

[8]

J. A. CañizoJ. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[9]

J. A. CarrilloR. ColomboP. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equations, 252 (2012), 3245-3277.  doi: 10.1016/j.jde.2011.11.003.  Google Scholar

[10]

J.A. CarrilloP. Gwiazda and A. Ulikowska, Splitting particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014), 2171-2197.  doi: 10.1142/S0218202514500183.  Google Scholar

[11]

R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753.  doi: 10.3934/dcds.2009.23.733.  Google Scholar

[12]

G. H. Cottet and P. A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal., 21 (1984), 52-76.  doi: 10.1137/0721003.  Google Scholar

[13]

A. M. de Roos, Numerical methods for structured population models: The escalator boxcar train, Numerical Methods for Partial Differential Equations, 4 (1988), 173-195.  doi: 10.1002/num.1690040303.  Google Scholar

[14]

A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Monographs in Population Biology 51, Princeton University Press, Princeton, 2013. Google Scholar

[15]

O. Diekmann and Ph. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Differential Equations, 215 (2005), 268-319.  doi: 10.1016/j.jde.2004.10.025.  Google Scholar

[16]

M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006). Google Scholar

[17]

J. EversS. Hille and A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015), 1068-1097.  doi: 10.1016/j.jde.2015.02.037.  Google Scholar

[18]

K. Ganguly and H. D. Victory Jr, On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989), 249-288.  doi: 10.1137/0726015.  Google Scholar

[19]

J. GoodmanT. Y. Hou and J. Lowengrub, Convergence of the point vortex method for the 2-D Euler equations, Comm. Pure Appl. Math., 43 (1990), 415-430.  doi: 10.1002/cpa.3160430305.  Google Scholar

[20]

P. GwiazdaJ. JabƚońskiA. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014), 1797-1820.  doi: 10.1002/num.21879.  Google Scholar

[21]

P. GwiazdaG. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM Journal on Mathematical Analysis, 44 (2012), 1103-1133.  doi: 10.1137/11083294X.  Google Scholar

[22]

P. GwiazdaT. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.  doi: 10.1016/j.jde.2010.02.010.  Google Scholar

[23]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010), 733-773.  doi: 10.1142/S021989161000227X.  Google Scholar

[24]

F. H. Harlow, The particle-in-cell computing method for fluid dynamics, Methods in computational physics, 3 (1964), 319-343.   Google Scholar

[25]

D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1996), 2099-2119.  doi: 10.1137/S0036142994266856.  Google Scholar

[26]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.  doi: 10.1007/s00205-013-0669-x.  Google Scholar

[27]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[28]

P. A. Raviart, An analysis of particle methods, Numerical Methods in Fluid Dynamics, Lecture Notes in Math. , Springer, Berlin, 1127 (1985), 243-324. doi: 10.1007/BFb0074532.  Google Scholar

[29]

E. Tadmor, A review of numerical methods for nonlinear partial differential equations, Bulletin of the American Mathematical Society, 49 (2012), 507-554.  doi: 10.1090/S0273-0979-2012-01379-4.  Google Scholar

[30]

C. Villani, Topics in Optimal Transportation, volume 58 of Graduate studies in mathematics, American Mathematical Society, 2003. doi: 10.1007/b12016.  Google Scholar

[31]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985.  Google Scholar

[32]

M. Westdickenberg and J. Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations, M2AN Math. Model. Numer. Anal., 44 (2010), 133-166.  doi: 10.1051/m2an/2009043.  Google Scholar

Figure 1.  Function $f$ and its Finite Range Approximation
Figure 2.  Function $f^{\varepsilon}$ and its approximation $\bar f^{\varepsilon}$
Figure 3.  Order of convergence

Results for $\varepsilon \in \{0.1, 0.0125, 0.0015625, \Delta t\}$ presented in Tables 1-4

Table 1.  The error of the FRA method for $\varepsilon=0.1$
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.1)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $0.0008865973$ -
$5.0000\cdot 10^{-2}$ $0.0005886514$ 0.59086543
$2.5000\cdot 10^{-2}$ $0.0003434678$ 0.77723882
$1.2500\cdot 10^{-2}$ $0.0002668756$ 0.36400752
$6.2500\cdot 10^{-3}$ $0.0002400045$ 0.15310595
$3.1250\cdot 10^{-3}$ $0.0002258416$ 0.08775007
$1.5625\cdot 10^{-3}$ $0.0002187803$ 0.04582869
$7.8125\cdot 10^{-4}$ $0.0002154824$ 0.02191237
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.1)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $0.0008865973$ -
$5.0000\cdot 10^{-2}$ $0.0005886514$ 0.59086543
$2.5000\cdot 10^{-2}$ $0.0003434678$ 0.77723882
$1.2500\cdot 10^{-2}$ $0.0002668756$ 0.36400752
$6.2500\cdot 10^{-3}$ $0.0002400045$ 0.15310595
$3.1250\cdot 10^{-3}$ $0.0002258416$ 0.08775007
$1.5625\cdot 10^{-3}$ $0.0002187803$ 0.04582869
$7.8125\cdot 10^{-4}$ $0.0002154824$ 0.02191237
Table 2.  The error of the FRA method for $\varepsilon=0.0125$
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0125)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$
$1.0000\cdot 10^{-1}$ $6.702793\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.255311\cdot 10^{-4}$ 0.6554979
$2.5000\cdot 10^{-2}$ $1.757573\cdot 10^{-4}$ 1.2756799
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 0.9135408
$6.2500\cdot 10^{-3}$ $5.658711\cdot 10^{-5}$ 0.7214983
$3.1250\cdot 10^{-3}$ $4.055238\cdot 10^{-5}$ 0.4806869
$1.5625\cdot 10^{-3}$ $3.332588\cdot 10^{-5}$ 0.2831438
$7.8125\cdot 10^{-4}$ $2.973577\cdot 10^{-5}$ 0.1644434
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0125)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$
$1.0000\cdot 10^{-1}$ $6.702793\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.255311\cdot 10^{-4}$ 0.6554979
$2.5000\cdot 10^{-2}$ $1.757573\cdot 10^{-4}$ 1.2756799
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 0.9135408
$6.2500\cdot 10^{-3}$ $5.658711\cdot 10^{-5}$ 0.7214983
$3.1250\cdot 10^{-3}$ $4.055238\cdot 10^{-5}$ 0.4806869
$1.5625\cdot 10^{-3}$ $3.332588\cdot 10^{-5}$ 0.2831438
$7.8125\cdot 10^{-4}$ $2.973577\cdot 10^{-5}$ 0.1644434
Table 3.  The error of the FRA method for $\varepsilon=0.0015625$
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0015625)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $6.616854\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.139056\cdot 10^{-4}$ 0.6768439
$2.5000\cdot 10^{-2}$ $1.570582\cdot 10^{-4}$ 1.3980025
$1.2500\cdot 10^{-2}$ $7.418803\cdot 10^{-5}$ 1.0820407
$6.2500\cdot 10^{-3}$ $3.649949\cdot 10^{-5}$ 1.0820407
$3.1250\cdot 10^{-3}$ $1.883518\cdot 10^{-5}$ 0.9544464
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 0.8669200
$7.8125\cdot 10^{-4}$ $6.277910\cdot 10^{-6}$ 0.7181537
$\Delta t$ ${\rm{Err}}(1,\Delta t,0.0015625)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $6.616854\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.139056\cdot 10^{-4}$ 0.6768439
$2.5000\cdot 10^{-2}$ $1.570582\cdot 10^{-4}$ 1.3980025
$1.2500\cdot 10^{-2}$ $7.418803\cdot 10^{-5}$ 1.0820407
$6.2500\cdot 10^{-3}$ $3.649949\cdot 10^{-5}$ 1.0820407
$3.1250\cdot 10^{-3}$ $1.883518\cdot 10^{-5}$ 0.9544464
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 0.8669200
$7.8125\cdot 10^{-4}$ $6.277910\cdot 10^{-6}$ 0.7181537
Table 4.  The error of the FRA method for $\varepsilon=\Delta t$
$\Delta t$ ${\rm{Err}}(1,\Delta t,\Delta t)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $8.865973\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.797352\cdot 10^{-4}$ 0.8860407
$2.5000\cdot 10^{-2}$ $1.991736\cdot 10^{-4}$ 1.2682122
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 1.0939823
$6.2500\cdot 10^{-3}$ $4.477800\cdot 10^{-5}$ 1.0591816
$3.1250\cdot 10^{-3}$ $2.162411\cdot 10^{-5}$ 1.0501496
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 1.0661308
$7.8125\cdot 10^{-4}$ $4.763338\cdot 10^{-6}$ 1.1164650
$\Delta t$ ${\rm{Err}}(1,\Delta t,\Delta t)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$ $8.865973\cdot 10^{-4}$ -
$5.0000\cdot 10^{-2}$ $4.797352\cdot 10^{-4}$ 0.8860407
$2.5000\cdot 10^{-2}$ $1.991736\cdot 10^{-4}$ 1.2682122
$1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 1.0939823
$6.2500\cdot 10^{-3}$ $4.477800\cdot 10^{-5}$ 1.0591816
$3.1250\cdot 10^{-3}$ $2.162411\cdot 10^{-5}$ 1.0501496
$1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 1.0661308
$7.8125\cdot 10^{-4}$ $4.763338\cdot 10^{-6}$ 1.1164650
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Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149

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