scholarly journals On asymptotic expansions for the fractional infinity Laplacian

2021 ◽  
pp. 1-16
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Marta Lewicka
Keyword(s):  
Asymptotic Expansions ◽  
Fractional Laplacian ◽  
Integral Type ◽  
Infinity Laplacian ◽  
Order Coefficient ◽  
Well Posed ◽  
Appl Math

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional ∞-Laplacian Δ ∞ s for s ∈ ( 1 2 , 1 ). This operator has been introduced and first studied in (Comm. Pure Appl. Math. 65 (2012) 337–380). Our expansions are parametrised by the radius of the removed singularity ε, and allow for the identification of Δ ∞ s ϕ ( x ) as the ε 2 s -order coefficient of the deviation of the ε-average from the value ϕ ( x ), in the limit ε → 0 +. The averages are well posed for functions ϕ that are only Borel regular and bounded.

A Survey in Mathematics for Industry: Two-timing and matched asymptotic expansions for singular perturbation problems

2011 ◽  
Vol 22 (6) ◽  
pp. 613-629 ◽  
Author(s):  
R. E. O'MALLEY ◽  
E. KIRKINIS
Keyword(s):  
Asymptotic Expansions ◽  
Multiple Scale ◽  
Singularly Perturbed ◽  
Matched Asymptotic Expansions ◽  
Amplitude Equations ◽  
Singularly Perturbed Problems ◽  
Multi Scale ◽  
Scale Method ◽  
Perturbation Problems ◽  
Appl Math

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems.Stud. Appl. Math.124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

scholarly journals Exact Green's formula for the fractional Laplacian and perturbations

2020 ◽  
Vol 126 (3) ◽  
pp. 568-592
Author(s):  
Gerd Grubb
Keyword(s):  
Fractional Laplacian ◽  
Order Term ◽  
Preceding Paper ◽  
Normal Derivative ◽  
Order Differential Operator ◽  
Nonlocal Term ◽  
First Order ◽  
Green’S Formula ◽  
Green's Formula ◽  
Well Posed

Let Ω be an open, smooth, bounded subset of $ \mathbb{R}^n $. In connection with the fractional Laplacian $(-\Delta )^a$ ($a>0$), and more generally for a $2a$-order classical pseudodifferential operator (ψdo) $P$ with even symbol, one can define the Dirichlet value $\gamma _0^{a-1}u$, resp. Neumann value $\gamma _1^{a-1}u$ of $u(x)$, as the trace, resp. normal derivative, of $u/d^{a-1}$ on $\partial \Omega $, where $d(x)$ is the distance from $x\in \Omega $ to $\partial \Omega $; they define well-posed boundary value problems for $P$. A Green's formula was shown in a preceding paper, containing a generally nonlocal term $(B\gamma _0^{a-1}u,\gamma _0^{a-1}v)_{\partial \Omega }$, where $B$ is a first-order ψdo on $\partial \Omega $. Presently, we determine $B$ from $L$ in the case $P=L^a$, where $L$ is a strongly elliptic second-order differential operator. A particular result is that $B=0$ when $L=-\Delta $, and that $B$ is multiplication by a function (is local) when $L$ equals $-\Delta $ plus a first-order term. In cases of more general $L$, $B$ can be nonlocal.

scholarly journals A generalization of Mortici lemma

2012 ◽  
Vol 21 (2) ◽  
pp. 129-134
Author(s):  
VASILE BERINDE ◽  
Keyword(s):  
Gamma Function ◽  
Asymptotic Expansions ◽  
Asymptotic Integration ◽  
Stirling Formula ◽  
Monotonic Functions ◽  
Factorial Function ◽  
Approximation Formulas ◽  
Appl Math ◽  
Best Estimates

The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.

A Diffusively Corrected Multiclass Lighthill-Whitham-Richards Traffic Model with Anticipation Lengths and Reaction Times

2013 ◽  
Vol 5 (05) ◽  
pp. 728-758 ◽  
Author(s):  
Raimund Bürger ◽  
Pep Mulet ◽  
Luis M. Villada
Keyword(s):  
Conservation Laws ◽  
Reaction Times ◽  
Cauchy Problems ◽  
Traffic Models ◽  
First Order ◽  
Systems Of Conservation Laws ◽  
The Stability ◽  
Well Posed ◽  
Appl Math ◽  
Nonlinear Nature

AbstractMulticlass Lighthill-Whitham-Richards traffic models [Benzoni-Gavage and Colombo, Euro. J. Appl. Math., 14 (2003), pp. 587–612; Wong and Wong, Transp. Res. A, 36 (2002), pp. 827–841] give rise to first-order systems of conservation laws that are hyperbolic under usual conditions, so that their associated Cauchy problems are well-posed. Anticipation lengths and reaction times can be incorporated into these models by adding certain conservative second-order terms to these first-order conservation laws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature.

Sign in / Sign up

Export Citation Format

Share Document