In this paper, we discuss numerical methods for fractional order problems. Some nonstandard finite difference schemes are presented and investigated. The application in the simulation of a fractional-order Brusselator system is hence presented. By means of some numerical experiments, we show the effectiveness of the proposed approach.
a b s t r a c tIn this paper, the Laplace Adomian Decomposition Method is implemented to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for HIV infection of CD4 + T cells. The technique is described and illustrated with numerical example. Some plots are presented to show the reliability and simplicity of the methods.
In this paper, we present a Lotka-Volterra predator-prey model with Allee effect. This system with general functional response has an Allee effect on prey population. A nonstandard finite difference scheme is constructed to transform the continuous time predator-prey model with Allee effect into the discrete time model. We use the Schur-Cohn criteria which deal with coefficients of the characteristic polynomial for determining the stability of discrete time system. The proposed numerical schemes preserve the positivity of the solutions with positive initial conditions. The new discrete-time model shows dynamic consistency with continuous-time model.
Key words Non-selfadjoint singular Schrödinger problem, spectral parameter in boundary condition, maximal dissipative operator, self-adjoint dilation, functional model, characteristic function, completeness of the system of eigenvectors and associated vector MSC (2000) Primary: 34B05, 34B40, 34L10; Secondary: 47A20, 47A40, 47A45, 47B44In this paper we consider a dissipative Schrödinger boundary value problem in the limit-circle case with the spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analyzes of this operator is adequate for the boundary value problem to be solved. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrödinger equation. We prove theorems on the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and the Schrödinger boundary value problem.
<abstract><p>Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.</p></abstract>
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