scholarly journals Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1439 ◽  
Author(s):  
Yu Huang ◽  
Mohammad Hadi Noori Skandari ◽  
Fatemeh Mohammadizadeh ◽  
Hojjat Ahsani Tehrani ◽  
Svetlin Georgiev Georgiev ◽  
...  
Keyword(s):  
Collocation Method ◽  
Numerical Approach ◽  
Space Time ◽  
Dirichlet Boundary Conditions ◽  
Spectral Collocation Method ◽  
Test Problems ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Burgers Equations ◽  
Chebyshev Spectral Collocation

This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

Vibration characteristics of multiple functionally graded nonuniform beams

2020 ◽  
pp. 107754632095676
Author(s):  
Ma’en S Sari ◽  
Sameer Al-Dahidi
Keyword(s):  
Collocation Method ◽  
Axial Direction ◽  
Functionally Graded ◽  
Spectral Collocation Method ◽  
Transverse Motion ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Engineering Structures ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Collocation Method

Based on the Euler–Bernoulli beam theory, the natural vibration behavior of functionally graded nonuniform multiple beams has been investigated. It is assumed that the beams are joined by elastic translational springs, and the properties of the beams vary along the axial direction. The Chebyshev spectral collocation method has been used to convert the governing differential equations of transverse motion into a system of algebraic equations that are put in the matrix–vector form. Then, the dimensionless transverse frequencies are obtained by solving the eigenvalue problem. The influence of several factors, such as the stiffness parameters of the coupling translational springs, the properties of the cross section of the beams, and the boundary conditions on the frequencies, has been carried out. The results generated from the Chebyshev spectral collocation method have been verified by comparing them with those reported in other studies and references from the literature. Several numerical examples have been presented and discussed to analyze the system under consideration. The authors hope that the findings of the current study are helpful in designing and characterizing multiple nonuniform thin engineering structures.

A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations

2020 ◽  
Vol 21 (7-8) ◽  
pp. 683-691
Author(s):  
Phumlani G. Dlamini ◽  
Vusi M. Magagula
Keyword(s):  
Collocation Method ◽  
Three Dimensional ◽  
Linearization Method ◽  
High Accuracy ◽  
Time Variable ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Burgers Equations ◽  
Grid Points ◽  
Linearization Techniques

AbstractIn this paper, we introduce the multi-variate spectral quasi-linearization method which is an extension of the previously reported bivariate spectral quasi-linearization method. The method is a combination of quasi-linearization techniques and the spectral collocation method to solve three-dimensional partial differential equations. We test its applicability on the (2 + 1) dimensional Burgers’ equations. We apply the spectral collocation method to discretize both space variables as well as the time variable. This results in high accuracy in both space and time. Numerical results are compared with known exact solutions as well as results from other papers to confirm the accuracy and efficiency of the method. The results show that the method produces highly accurate solutions and is very efficient for (2 + 1) dimensional PDEs. The efficiency is due to the fact that only few grid points are required to archive high accuracy. The results are portrayed in tables and graphs.

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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