Gao et al. [12] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O(k 3−α), 0 < α < 1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [22] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [22] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O(k 3−α), 0 < α < 1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O(k 3−α), 0 < α < 1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O(k 3−α), 0 < α < 1 as in Gao et al. [12] (2014) by approximating the Hadamard finitepart integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O(k 3−α), 0 < α < 1 for any fixed t n > 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
This paper proposes a malaria transmission model to describe the dynamics of malaria transmission in the human and mosquito populations. This model emphasizes the impact of limited resource on malaria transmission. We derive a formula for the basic reproductive number of infection and investigate the existence of endemic equilibria. It is shown that this model may undergo backward bifurcation, where the locally stable disease-free equilibrium co-exists with an endemic equilibrium. Furthermore, we determine conditions under which the disease-free equilibrium of the model is globally asymptotically stable. The global stability of the endemic equilibrium is also studied when the basic reproductive number is greater than one. Finally, numerical simulations to illustrate our findings and brief discussions are provided.
Dengue fever is caused by dengue virus and transmitted by Aedes mosquitoes. A promising avenue to control this disease is to infect the wild Aedes population with the bacterium Wolbachia driven by cytoplasmic incompatibility (CI). To study the invasion of Wolbachia into wild mosquito population, we formulate a discrete competition model and analyze the competition between released mosquitoes and wild mosquitoes. We show the global asymptotic properties of the trivial equilibrium, boundary equilibrium, and positive equilibrium and give the conditions for the successful invasion of Wolbachia. Finally, we verify our findings by numerical simulations.
In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray-Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.
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