Abstract
The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.
Highlights
The diffusion-convection-reaction (DCR) equation arises in a number of physical phenomena, such as, the dispersion of chemicals in reactors,[1] the shocks in hydrodynamical flows,[2] the mass transfer in the capillary membrane bioreactor[3] and the tracer dispersion in the porous medium.[4]
The dominated dynamic characteristics of the DCR equation can be successfully preserved by many numerical methods, such as, finite element method (FEM),[6] finite-difference time-domain (FDTD) method[7] and finite-volume time-domain (FVTD) method.[8]
The explicit eigenvalues and eigenvectors are derived by analyzing the relationship between the spatial differential operators and boundary conditions
Summary
The diffusion-convection-reaction (DCR) equation arises in a number of physical phenomena, such as, the dispersion of chemicals in reactors,[1] the shocks in hydrodynamical flows,[2] the mass transfer in the capillary membrane bioreactor[3] and the tracer dispersion in the porous medium.[4]. The method has been widely applied in a class of models with the first or second order spatial differential operators, such as, the aluminum alloy hot rolling process,[11] snap curing process,[12] fixed-bed reactor process[13] and microwave heating process.[14] it is difficult to directly derive the spatial function for the PDEs with the mixed spatial differential operator. The model-based method, which can directly derive eigenfunctions, needs to be explored for the dimensionality reduction problem of the DCR equation.
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