Who Solved the Bernoulli Differential Equation and How Did They Do It?

Parker, Adam E.

doi:10.4169/college.math.j.44.2.089

Cited by 11 publications

(5 citation statements)

References 3 publications

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“…Proof Based on Bernoulli's equation in Reference 38, from (4), one gets

{\dot{x}}_k(t)=\hslash (t){x}_k(t)-\left({c}_1\sum \limits_{j=1}^M\kern.15em {a}_{kj}+{\gamma}_k\right){x}_k^2(t),

where

\hslash (t)={c}_1\sum \limits_{j=1}^M\kern.15em {a}_{kj}{x}_j(t)+{c}_1\sum \limits_{j=1}^M\kern.15em {l}_{kj}{x}_{\vartheta_j}+{\gamma}_k{x}_{\vartheta_j}

. Let

{z}_k(t)={x}_k^{-1}(t)

.…”

Section: Resultsmentioning

confidence: 99%

Group consensus of multi‐agent systems with reference states via pinning control

Wu,

Sun,

Han

et al. 2023

Intl J Robust & Nonlinear

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This article investigates the group consensus via pinning control for continuous‐time first‐order and second‐order multi‐agent systems (MASs) with reference states. For the group consensus of first‐order MASs, the dependence between the agent's state and the control input is considered. For second‐order MASs, group consensus control without the velocity information of agents is considered. Instead, the virtual velocity estimation controller is designed. Meanwhile, for the designed control protocols, not only under fixed topology, but also under switching topology are considered. It is demonstrated that group consensus could be obtained under the proposed control protocols by using graph theory and stability theory. Finally, a series of numerical examples are provided to verify the control performance of the propounded control protocols.

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“…Proof Based on Bernoulli's equation in Reference 38, from (4), one gets

{\dot{x}}_k(t)=\hslash (t){x}_k(t)-\left({c}_1\sum \limits_{j=1}^M\kern.15em {a}_{kj}+{\gamma}_k\right){x}_k^2(t),

where

\hslash (t)={c}_1\sum \limits_{j=1}^M\kern.15em {a}_{kj}{x}_j(t)+{c}_1\sum \limits_{j=1}^M\kern.15em {l}_{kj}{x}_{\vartheta_j}+{\gamma}_k{x}_{\vartheta_j}

. Let

{z}_k(t)={x}_k^{-1}(t)

.…”

Section: Resultsmentioning

confidence: 99%

Group consensus of multi‐agent systems with reference states via pinning control

Wu,

Sun,

Han

et al. 2023

Intl J Robust & Nonlinear

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“…The equation ( 65) is the well known Bernoulli equation (e.g. see [15]). Under assumptions of corollary 1 the model was originally considered in [8].…”

Section: Remark On Fractional Sir Modelsmentioning

confidence: 99%

Discrete SIR model on a homogeneous tree and its continuous limit

Gairat¹,

Shcherbakov

2022

J. Phys. A: Math. Theor.

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We study a discrete Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to inﬁnity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to inﬁnity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution.

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“…Thus: In this study we introduced the q-analogue Bernoulli's equation. A free analogue of some nuclear algebras of operators acting on space of holomorphic functions on a free analogue complexification of real nuclear space can be studied and we expect to develop a new quantum white noise analogue of free-Bernoulli's equation (Rguigui, 2015;Parker, 2013;Rguigui, 2016ab;2018a-b;Altoum, 2018).…”

Section: S Equationmentioning

confidence: 99%