Nash equilibrium approximation of some class of stochastic differential games: A combined Chebyshev spectral collocation method with policy iteration

2019 ◽  
Vol 362 ◽  
pp. 41-54 ◽  
Author(s):  
Z. Nikooeinejad ◽  
M. Heydari
Keyword(s):  
Nash Equilibrium ◽  
Differential Games ◽  
Collocation Method ◽  
Stochastic Differential Games ◽  
Policy Iteration ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Collocation Method

scholarly journals Chebyshev Spectral Collocation Method for Population Balance Equation in Crystallization

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 317
Author(s):  
Chunlei Ruan
Keyword(s):  
Collocation Method ◽  
Spectral Collocation Method ◽  
Population Balance Equation ◽  
Spectral Collocation ◽  
Numerical Examples ◽  
Size Dependent ◽  
Chebyshev Spectral Collocation ◽  
Diffusive Term ◽  
Dependent Growth ◽  
Chebyshev Spectral Collocation Method

The population balance equation (PBE) is the main governing equation for modeling dynamic crystallization behavior. In the view of mathematics, PBE is a convection–reaction equation whose strong hyperbolic property may challenge numerical methods. In order to weaken the hyperbolic property of PBE, a diffusive term was added in this work. Here, the Chebyshev spectral collocation method was introduced to solve the PBE and to achieve accurate crystal size distribution (CSD). Three numerical examples are presented, namely size-independent growth, size-dependent growth in a batch process, and with nucleation, and size-dependent growth in a continuous process. Through comparing the results with the numerical results obtained via the second-order upwind method and the HR-van method, the high accuracy of Chebyshev spectral collocation method was proven. Moreover, the diffusive term is also discussed in three numerical examples. The results show that, in the case of size-independent growth (PBE is a convection equation), the diffusive term should be added, and the coefficient of the diffusive term is recommended as 2G × 10−3 to G × 10−2, where G is the crystal growth rate.

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