scholarly journals On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 554-562
Author(s):  
Hillary Muzara ◽  
Stanford Shateyi ◽  
Gerald Tendayi Marewo
Keyword(s):  
Collocation Method ◽  
Linearization Method ◽  
Spectral Collocation Method ◽  
Two Dimensional ◽  
Weighted Residual Method ◽  
Spectral Collocation ◽  
Finite Differences Method ◽  
Chebyshev Spectral Collocation ◽  
Bratu Problem ◽  
Chebyshev Spectral Collocation Method

AbstractIn this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

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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