Chebyshev spectral-collocation method for a class of weakly singular Volterra integral equations with proportional delay

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
Z. Gu ◽  
Y. Chen
Keyword(s):  
Error Analysis ◽  
Integral Equations ◽  
Collocation Method ◽  
Volterra Integral Equations ◽  
Spectral Collocation Method ◽  
Proportional Delay ◽  
Spectral Collocation ◽  
Weakly Singular ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Collocation Method

Abstract-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

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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