2016
DOI: 10.1016/j.jmaa.2016.06.047
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Weak asymptotic methods for scalar equations and systems

Abstract: Please cite this article in press as: E. Abreu et al., Weak asymptotic methods for scalar equations and systems, J. Math. Anal. Appl. (2016), http://dx. AbstractIn this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslov asymptotic analysis. We obtain a sequence of functions which tend to satisfy t… Show more

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Cited by 27 publications
(45 citation statements)
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References 65 publications
(149 reference statements)
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“…We consider a novel Lagrangian-Eulerian formulation that can be viewed as an extension of previous works [2,3,8] for practical construction of numerical solutions for balance law problems, but following innovative recent ideas introduced in [1] to construct weak asymptotic methods for scalar equations and systems of conservation law equations. For simplicity, we consider the particular scalar equation with u = u(x, t)…”
Section: Methodsmentioning
confidence: 99%
See 1 more Smart Citation
“…We consider a novel Lagrangian-Eulerian formulation that can be viewed as an extension of previous works [2,3,8] for practical construction of numerical solutions for balance law problems, but following innovative recent ideas introduced in [1] to construct weak asymptotic methods for scalar equations and systems of conservation law equations. For simplicity, we consider the particular scalar equation with u = u(x, t)…”
Section: Methodsmentioning
confidence: 99%
“…We provide a formal development of the analogue Lagrangian-Eulerian scheme [4,5,8] for numerically solving the initial value problem (1). As in the Lagrangian-Eulerian schemes [3][4][5], a local mass balance equation is obtained by integrating the hyperbolic balance law (1) over the region in the space-time domain.…”
Section: Methodsmentioning
confidence: 99%
“…The Weak Asymptotics Method (see e.g. [6] - [15] and references therein) takes into account the fact that soliton-type solutions which are smooth for ε > 0 become non-smooth in the limit as ε → 0. Thus, it is possible to treat such solutions as a mapping…”
Section: Main Definitionsmentioning
confidence: 99%
“…Obviously, there is not any hope to find both the exact multi-soliton solution to (1) and an asymptotics in the classical sense. So, we will construct a weak asymptotic solution.…”
Section: Main Definitionsmentioning
confidence: 99%
“…So, we will construct a weak asymptotic solution. The weak asymptotics method (see, e.g., [1][2][3][4][5][6][7], [9,10] and references therein) takes into account the fact that soliton-type solutions which are smooth for 0 > ε become non-smooth in the limit as . 0 → ε Thus, it is possible to treat such solutions as a mapping…”
Section: Main Definitionsmentioning
confidence: 99%