Simultaneous and sequential solution of a stiff differential equation system

1978 ◽  
Vol 25 (2) ◽  
pp. 207-218 ◽  
Author(s):  
S. M. Roberts
Keyword(s):  
Differential Equation ◽  
Equation System ◽  
Differential Equation System ◽  
Stiff Differential Equation ◽  
Sequential Solution

scholarly journals Transient Dynamics in the Random Growth and Reset Model

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 306
Author(s):  
Tamás S. Biró ◽  
Lehel Csillag ◽  
Zoltán Néda
Keyword(s):  
Differential Equation ◽  
Mean Field ◽  
Equation System ◽  
Transient Dynamics ◽  
Type Model ◽  
Discrete Version ◽  
Differential Equation System ◽  
Practical Applications ◽  
Random Growth ◽  
Recursive Integration

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain; however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.

scholarly journals Kestabilan Model Epidemi SEIR Dengan Laju Insidensi

2015 ◽  
Vol 10 (2) ◽  
pp. 74
Author(s):  
Roni Tri Putra ◽  
Sukatik - ◽  
Sri Nita
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Equation System ◽  
Differential Equation System ◽  
Equilibrium Existence

In this paper, it will be studied stability for a SEIR epidemic model with infectious force in latent, infected and immune period with incidence rate. From the model it will be found investigated the existence and uniqueness solution  of points its equilibrium. Existence solution of points equilibrium proved by show its differential equations system of equilibrium continue, and uniqueness solution of points equilibrium proved by show its differential equation system of equilibrium differentiable continue. 

scholarly journals An accelerated differential equation system for generalized equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qiyuan Wei ◽  
Liwei Zhang
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Maximal Monotone ◽  
Equation System ◽  
Generalized Equation ◽  
Generalized Equations ◽  
Differential Equation System ◽  
Step Size ◽  
Yosida Regularization ◽  
Numerical Discretization

<p style='text-indent:20px;'>An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>to the solution set <inline-formula><tex-math id="M2">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> of a generalized equation <inline-formula><tex-math id="M3">\begin{document}$ 0 \in T(x) $\end{document}</tex-math></inline-formula>. With smart choices of parameters <inline-formula><tex-math id="M4">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \alpha(t) $\end{document}</tex-math></inline-formula>, we prove the weak convergence of the trajectory to some point of <inline-formula><tex-math id="M6">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ \|\dot{x}(t)\|\leq {\rm O}(1/t) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ t\rightarrow +\infty $\end{document}</tex-math></inline-formula>. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory <inline-formula><tex-math id="M9">\begin{document}$ x^{h}(t) \rightarrow x(t) $\end{document}</tex-math></inline-formula> on interval <inline-formula><tex-math id="M10">\begin{document}$ [0,+\infty) $\end{document}</tex-math></inline-formula> is demonstrated when the step size <inline-formula><tex-math id="M11">\begin{document}$ h \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Theoretical Vibration Analysis Regarding Excitation due to Elliptical Shaft Journals in Sleeve Bearings of Electrical Motors

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Ulrich Werner
Keyword(s):  
Differential Equation ◽  
Mathematical Description ◽  
Vibration Analysis ◽  
Equation System ◽  
Electrical Machines ◽  
Differential Equation System ◽  
Kinematic Constraints ◽  
Linear Differential ◽  
Electrical Motors ◽  
Rotor Mass

This paper shows a theoretical vibration analysis regarding excitation due to elliptical shaft journals in sleeve bearings of electrical motors, based on a simplified rotordynamic model. It is shown that elliptical shaft journals lead to kinematic constraints regarding the movement of the shaft journals on the oil film of the sleeve bearings and therefore to an excitation of the rotordynamic system. The solution of the linear differential equation system leads to the mathematical description of the movement of the rotor mass, the shaft journals, and the sleeve bearing housings. Additionally the relative movements between the shaft journals and the bearing housings are deduced, as well as the bearing housing vibration velocities. The presented simplified rotordynamic model can also be applied to rotating machines, other than electrical machines. In this case, only the electromagnetic spring valuecmhas to be put to zero.

A Short Method to Compute Nusselt Numbers in Rectangular and Annular Channels With any Ratio of Constant Heat Rate

2010 ◽  
Author(s):  
Alexandre Malon ◽  
Thierry Muller
Keyword(s):  
Differential Equation ◽  
Nusselt Number ◽  
Heat Rate ◽  
Equation System ◽  
Radius Ratio ◽  
Differential Equation System ◽  
Simple Method ◽  
Nusselt Numbers ◽  
Annular Channels ◽  
Wide Range

An analytic investigation of the thermal exchanges in channels is conducted with the prospect of building a simple method to determine the Nusselt number in steady, laminar or turbulent and monodimensional flow through rectangular and annular spaces with any ratio of constant and uniform heat rate. The study of the laminar case leads to explicit laws for the Nusselt number while the turbulent case is solved using a Reichardt turbulent viscosity model resulting in easy to solve one-dimensional ordinary differential equation system. This differential equation system is solved using a Matlab based boundary value problems solver (bvp4c). A wide range of Reynolds, Prandtl and radius ratio is explored with the prospect of building correlation laws allowing the computing of Nusselt numbers for any radius ratio. Those correlations are in good agreement with the results obtained by W.M. Kays and E.Y. Leung in 1963 [1]. They conduced a similar analysis but with an experimental basis, they explored a greater range of Prandtl but only a few discreet radius ratio. The correlations are also compared with a CFD analysis made on a case extracted from the Re´acteur Jules Horowitz.

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