The Boundary Function Method for Singular Perturbation Problems

Vasil’eva, A. B.; Butuzov, V. F.; Kalachev, Leonid

doi:10.1137/1.9781611970784

Cited by 284 publications

(217 citation statements)

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“…For any initial value, the boundary layer system (14) and the boundary layer equation (15) have unique global solutions, which is ensured by the global existence theorem from Sec. 2.…”

Section: Theorem 3 If Assumptions 5-7 Are Fulfilled Then the Solutimentioning

confidence: 90%

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Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Tafintseva¹,

Machina²,

Ponosov³

2013

Nonlinear Analysis: Real World Applications

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We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.

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“…For any initial value, the boundary layer system (14) and the boundary layer equation (15) have unique global solutions, which is ensured by the global existence theorem from Sec. 2.…”

Section: Theorem 3 If Assumptions 5-7 Are Fulfilled Then the Solutimentioning

confidence: 90%

“…It is sufficient for our purposes to apply the classical result on singular perturbations, known in the literature as Tikhonov's theorem (see e.g. [14]). To do it, we need the following assumptions: Assumption 5.…”

Section: Singular Perturbation Analysis In Codimensionmentioning

confidence: 99%

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Tafintseva¹,

Machina²,

Ponosov³

2013

Nonlinear Analysis: Real World Applications

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“…As we mentioned above, the solution of the degenerate system ̅( ) does not satisfy initial condition in (4) and, therefore, it may be assumed the existence of a boundary layer structure in the solution as → 0. Following [6][7][8][9], in accordance with the method of boundary functions [10,11], such a solution can be found as a sum of a regular and boundary-layer series…”

Section: Asymptotic Expansionsmentioning

confidence: 99%

“…Terms of higher order approximations can be found from linear equations. The asymptotic character of the expansions is justified as described [6][7][8][9][10][11].…”

mentioning

confidence: 99%

Reduction in Initial Boundary Value Problem for Hiv Evolution Model

Арчибасов¹

2016

Collection of Selected Papers of the II International Conference on Information Technology and Nanotechnology

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Abstract. HIV evolution model that describes the dynamics of concentrations of uninfected and infected cells is considered in the paper. The introduction of dimensionless variables and parameters leads to the initial boundary value problem for singularly perturbed system of partial integro-differential equations. By Tikhonov-Vasil'eva boundary functions method asymptotic solution with boundary layer is constructed. Numeric simulation of complete and reduced systems are also given.

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“…We use the boundary function method (see [6], [15]) to solve this singular perturbation problem (3.9). The perturbation term of the induction is decomposed as follows for all integer N Notation: The superscript s, out (resp.…”

Section: 21 Asymptotic Expansionsmentioning

confidence: 99%

Asymptotic analysis of magnetic induction with high frequency for solid conductors

Coulaud¹

1998

ESAIM: M2AN

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The Boundary Function Method for Singular Perturbation Problems

Cited by 284 publications

References 0 publications

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Reduction in Initial Boundary Value Problem for Hiv Evolution Model

Asymptotic analysis of magnetic induction with high frequency for solid conductors

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