Homogeneization problems for ordinary differential equations

Piccinini, Livio Clemente

doi:10.1007/bf02843869

Cited by 15 publications

(14 citation statements)

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“…Let γ be the rotation number (see section 1.3) and assume that a " p1, γq satisfies (13) with some constants C ą 0 and κ ą 0 such that k ą 3`κ. Then there is a linear function X 0 ptq " p`Bt, B P R 2 such that (15) |X ε ptq´X 0 ptq| ďĈε, t P r0, 8q…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

On a conjecture of De Giorgi related to homogenization

Karakhanyan

Shahgholian

2017

Annali di Matematica

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For a periodic vector field F, let X ε solve the dynamical systemIn (Set Valued Anal 2(1-2):175-182, 1994) Ennio De Giorgi enquiers whether from the existence of the limit X 0 (t) := lim ε→0 X ε (t) one can conclude that dX 0 dt = constant. Our main result settles this conjecture under fairly general assumptions on F, which in some cases may also depend on t-variable. Once the above problem is solved, one can apply the result to the corresponding transport equation, in a standard way. This is also touched upon in the text to follow.

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Section: Preliminaries and Main Resultsmentioning

confidence: 99%

On a conjecture of De Giorgi related to homogenization

Karakhanyan

Shahgholian

2017

Annali di Matematica

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“…Homogenization problems for ODEs were studied by [11] but it is worth pointing out that our particular case does not fall into the theory described in [11]. Fortunately the specific structure of the transport equation allows us to do a complete analysis of the problem and even to compute explicitly the averaged equation.…”

Section: Modelmentioning

confidence: 99%

Mathematical modeling of transport and degradation of feedstuffs in the small intestine

Taghipoor

Lescoat

Licois

et al. 2012

Journal of Theoretical Biology

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We describe a mathematical model of digestion in the small intestine. The main interest of our work is to consider simultaneously the different aspects of digestion i.e. transport of the bolus all along the intestine, feedstuffs degradation according to the enzymes and local physical conditions, and nutrients absorption. A system of coupled ordinary differential equations is used to model these phenomena. The major unknowns of this system are the position of the bolus and its composition. This system of equations is solved numerically. We present several numerical computations for the degradation, absorption and transport of the bolus with acceptable accuracy regarding the overall behavior of the model and also when challenged versus experimental data. The main feature and interest of this model are its genericity. Even if we are at an early stage of development, our approach can be adapted to deal with contrasted feedstuffs in nonruminant animal to predict the composition and velocity of bolus in the small intestine.

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“…Let us mention that, for some specific functions f , explicit formulas for f can be obtained (see for instance [23], and the examples below). Even if f may not be Lipschitz continuous in u, we can show the existence and uniqueness of the solution of (1.6), taking advantage of the monotonicity of f (u 0 , t) in u 0 .…”

Section: 3)mentioning

confidence: 99%

“…We can also cite the work in [8] about the rate of convergence in periodic homogenization of first-order stationary Hamilton-Jacobi equations, where an error estimate in ǫ 1/3 is obtained for Hamilton-Jacobi equations with Lipschitz effective Hamiltonian. For the problems of homogenization of ODEs, we refer the reader to [23,24]. We also refer the reader to [2,9,10,19,21,22,26] for problems on homogenization of nonlinear first-order ODEs and/or the associated linear transport equations.…”

Section: Brief Review Of the Literaturementioning

confidence: 99%

On the Rate of Convergence in Periodic Homogenization of Scalar First-Order Ordinary Differential Equations

Ibrahim¹,

Monneau²

2010

SIAM J. Math. Anal.

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In this paper, we study the rate of convergence in periodic homogenization of scalar ordinary differential equations. We provide a quantitative error estimate between the solutions of a first-order ordinary differential equation with rapidly oscillating coefficients and the limiting homogenized solution. As an application of our result, we obtain an error estimate for the solution of some particular linear transport equations.

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Homogeneization problems for ordinary differential equations

Cited by 15 publications

References 0 publications

On a conjecture of De Giorgi related to homogenization

On a conjecture of De Giorgi related to homogenization

Mathematical modeling of transport and degradation of feedstuffs in the small intestine

On the Rate of Convergence in Periodic Homogenization of Scalar First-Order Ordinary Differential Equations

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