Abstract
We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.
Highlights
Logistic equation with space distribution is an important mathematical model in describing space population ecology with carrying capacity, which is given by Brauer & Castillo-Chavez (2012)ut = ∇ · (D∇u) + ru(1 − u/K), (1.1)where u = u(x, t) denotes the population density at t > 0 and x ∈ Ω, Ω is a bounded domain with smooth boundary, and D is the diffusion coefficient(tensor), r > 0 is the intrinsic growth rate, and K > 0 is the carrying capacity of the population
There are a lot of researches on nonlinear reaction-diffusion systems like Eq(1.1), and most of the studies focus on dynamics analysis from the viewpoints of population ecology and epidemiology, see Bai & Wang (2005), Balanov et al (2008), Chakraborty et al (2007), Jin & Zhao (2008), Korobenko & Braverman (2009), Wang et al (2018), Zhao (2003) for instance
We refer to El-Nabulsi (2011, 2019, 2020), Lima et al (2007), Magin (2006), Moshrefi-Torbati & Hammond (1998), Singla & Gupta (2016) for the applications of fractional calculus in quantum mechanics, molecular physics, fluid and microfilaments, bioengineering, signal processing and system control, etc
Summary
Logistic equation with space distribution is an important mathematical model in describing space population ecology with carrying capacity, which is given by Brauer & Castillo-Chavez (2012). In the case of the diffusion coefficient and the initial distribution are both constants, the unique existence of weak solution to the time-fractional diffusion equation with a nonlinear term is proved by Luchko et al (2013), and most of the researches on nonlinear fractional differential equations are concerned with numerical methods, see Liu et al (2015, 2016, 2018), Li & Rui (2017) for instance. A two-grids algorithm based on finite difference method was utilized to solve a 2D nonlinear time fractional mobile-immobile solute transport model, and the L2 stability and convergence were proved (Qiu et al, 2020), and a multi-grids algorithm was set forth for solving nonlinear fractional diffusion equations in 1D and 2D cases, and the unique solvability, stability and convergence of the numerical schemes were proved by the optimal error estimates (Maurya et al, 2021).
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