This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity ...This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.展开更多
The purpose of this paper is to devise a new model of heat conduction for a body which istotally insulated at its boundary. The essence of the new model is modification of the Fourier law insuch a way that the resulti...The purpose of this paper is to devise a new model of heat conduction for a body which istotally insulated at its boundary. The essence of the new model is modification of the Fourier law insuch a way that the resulting heat conduction problem will be fonnulated as an initial value problemwith a nonstandard partial differential equation. The values of normal derivatives of the heat fiuxto various orders for all tmies (t> 0) will be detemined by the governing partial differential equa-tion and the values of normal derivatives of fluxes at the initial time (t=0). We study the properitesof this model and demonstrate its validity for a simple onedimensional case of a thin bar insulatedat both ends.展开更多
文摘This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.
文摘The purpose of this paper is to devise a new model of heat conduction for a body which istotally insulated at its boundary. The essence of the new model is modification of the Fourier law insuch a way that the resulting heat conduction problem will be fonnulated as an initial value problemwith a nonstandard partial differential equation. The values of normal derivatives of the heat fiuxto various orders for all tmies (t> 0) will be detemined by the governing partial differential equa-tion and the values of normal derivatives of fluxes at the initial time (t=0). We study the properitesof this model and demonstrate its validity for a simple onedimensional case of a thin bar insulatedat both ends.
