The nonlinear fractional point reactor kinetics equation in the presence of Newtonian temperature reactivity feedback with a multi-group of delayed neutrons,which describes the spectrum behavior of neutron density int...The nonlinear fractional point reactor kinetics equation in the presence of Newtonian temperature reactivity feedback with a multi-group of delayed neutrons,which describes the spectrum behavior of neutron density into the homogenous nuclear reactors, is developed. This system is one of the most important stiff coupled nonlinear fractional differentials for nuclear reactor dynamics. The generalization of Taylor's formula that involves Caputo fractional derivatives is developed in an attempt to overcome the difficulty of the stiffness of the nonlinear fractional differential model. Moreover, the general fractional derivatives are calculated analytically throughout this work. Furthermore, the local and global estimated errors were analyzed, which suggest that the error quantification should take into account the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error(local truncation error). The neutron density response with time is analyzed for the anomalous diffusion, sub-diffusion, and super-diffusion processes.展开更多
In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one （Jumarie） has proposed rece...In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one （Jumarie） has proposed recently an alternative referred to as （local） modified Riemann-Liouville definition, which directly, provides a Taylor＇s series of fractional order for non differentiable functions. We examine here in which way this calculus can be used as a framework for a differential geometry of fractional or- der. One will examine successively implicit function, manifold, length of curves, radius of curvature, Christoffel coefficients, velocity, acceleration. One outlines the application of this framework to La- grange optimization in mechanics, and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.展开更多
文摘The nonlinear fractional point reactor kinetics equation in the presence of Newtonian temperature reactivity feedback with a multi-group of delayed neutrons,which describes the spectrum behavior of neutron density into the homogenous nuclear reactors, is developed. This system is one of the most important stiff coupled nonlinear fractional differentials for nuclear reactor dynamics. The generalization of Taylor's formula that involves Caputo fractional derivatives is developed in an attempt to overcome the difficulty of the stiffness of the nonlinear fractional differential model. Moreover, the general fractional derivatives are calculated analytically throughout this work. Furthermore, the local and global estimated errors were analyzed, which suggest that the error quantification should take into account the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error(local truncation error). The neutron density response with time is analyzed for the anomalous diffusion, sub-diffusion, and super-diffusion processes.
文摘In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one （Jumarie） has proposed recently an alternative referred to as （local） modified Riemann-Liouville definition, which directly, provides a Taylor＇s series of fractional order for non differentiable functions. We examine here in which way this calculus can be used as a framework for a differential geometry of fractional or- der. One will examine successively implicit function, manifold, length of curves, radius of curvature, Christoffel coefficients, velocity, acceleration. One outlines the application of this framework to La- grange optimization in mechanics, and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.
