Abstract
The theory of dynamical systems and their widespread applications involve, for example, the Lane–Emden-type equations which are known to arise in initial- and boundary-value problems with singularity at the time t=0. The main objective of this paper is to make use of some mathematical analytic tools and techniques in order to numerically solve some reaction–diffusion equations, which arise in spherical catalysts and spherical biocatalysts, by applying the Chebyshev spectral collocation method. The proposed scheme has good accuracy. The results are demonstrated by means of illustrative graphs and numerical tables. The accuracy of the proposed method is verified by a comparison with the results which are derived by using analytical methods.
Highlights
The Lane–Emden-type equations are known to arise in initial- and boundary-value problems with singularity at the time t = 0
We first show the effect of the parameters μ, α and ρ on the concentration of the substance in a spherical catalyst model
We show here the effect of the parameters μ, α and ρ on the concentration of the substance in a spherical biocatalyst model
Summary
The Lane–Emden-type equations are known to arise in initial- and boundary-value problems with singularity at the time t = 0. Analytical solutions in the neighbourhood of t = 0 are always possible to find (see [1,2]), such as those given in [3,4,5,6]: v (t) +. Equation (1) arises in astrophysics and many other branches of science such as physics, chemistry and bio-mathematics (see, for example, [7,8,9,10,11]). The Lane–Emden boundary value problem (BVP) models the chemical species dimensionless concentration within a spherical catalyst and is given in [12] as follows:
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