Solution of stokes flow problem using biharmonic equation formulation and multiquadrics method

Марданов, Р. Ф.; Zaripov, S. K.

doi:10.1134/s1995080216030161

Cited by 7 publications

(6 citation statements)

References 9 publications

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“…We also recall that fourth order elliptic operators in R d in particular, the biharmonic operator, play also a central role in a wide class of physical models such as linear elasticity theory, rigidity problems (for instance, construction of suspension bridges) and in streamfunction formulation of Stokes flows (see e.g. [9,25,27] and references therein). Moreover, recent investigations have shown that the Laplace and biharmonic operators have high potential in image compression with the optimized and sufficiently sparse stored data [15].…”

Section: Introductionmentioning

confidence: 99%

Expansion of eigenvalues of the perturbed discrete bilaplacian

2022

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We consider the family $$\begin{aligned} {\widehat{{ H}}}_\mu := {\widehat{\varDelta }} {\widehat{\varDelta }} - \mu {\widehat{{ V}}},\qquad \mu \in {\mathbb {R}}, \end{aligned}$$ H ^ μ : = Δ ^ Δ ^ - μ V ^ , μ ∈ R , of discrete Schrödinger-type operators in d-dimensional lattice $${\mathbb {Z}}^d$$ Z d , where $${\widehat{\varDelta }}$$ Δ ^ is the discrete Laplacian and $${\widehat{{ V}}}$$ V ^ is of rank-one. We prove that there exist coupling constant thresholds $$\mu _o,\mu ^o\ge 0$$ μ o , μ o ≥ 0 such that for any $$\mu \in [-\mu ^o,\mu _o]$$ μ ∈ [ - μ o , μ o ] the discrete spectrum of $${\widehat{{ H}_\mu }}$$ H μ ^ is empty and for any $$\mu \in {\mathbb {R}}\setminus [-\mu ^o,\mu _o]$$ μ ∈ R \ [ - μ o , μ o ] the discrete spectrum of $${\widehat{{ H}_\mu }}$$ H μ ^ is a singleton $$\{e(\mu )\},$$ { e ( μ ) } , and $$e(\mu )<0$$ e ( μ ) < 0 for $$\mu >\mu _o$$ μ > μ o and $$e(\mu )>4d^2$$ e ( μ ) > 4 d 2 for $$\mu <-\mu ^o.$$ μ < - μ o . Moreover, we study the asymptotics of $$e(\mu )$$ e ( μ ) as $$\mu \searrow \mu _o$$ μ ↘ μ o and $$\mu \nearrow -\mu ^o$$ μ ↗ - μ o as well as $$\mu \rightarrow \pm \infty .$$ μ → ± ∞ . The asymptotics highly depends on d and $${\widehat{{ V}}}.$$ V ^ .

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Section: Introductionmentioning

confidence: 99%

Expansion of eigenvalues of the perturbed discrete bilaplacian

2022

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“…[MW87] and the references therein. See also formula (1) in [MZ16] and the references therein for other classical applications of the biharmonic operator in the study of steady state incompressible fluid flows at small Reynolds numbers under the Stokes flow approximation assumption. In our framework, we will present a simple game-theoretical model for the problem in (1.1) in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

2019

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We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models.We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in [DKV18], and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem.We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter.Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.

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“…Fourth order elliptic operators in R d in particular, the biharmonic operator, play a central role in a wide class of physical models such as linear elasticity theory, rigidity problems (for instance, construction of suspension bridges) and in streamfunction formulation of Stokes flows (see e.g. [7,22,24] and references therein). Moreover, recent investigations have shown that the Laplace and biharmonic operators have high potential in image compression with the optimized and sufficiently sparse stored data [13].…”

Section: Introductionmentioning

confidence: 99%

Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian

Khalkhuzhaev,

Kholmatov,

Pardabaev

2019

Preprint

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We consider the family hµ := ∆ ∆ − µ v, µ ∈ R, of discrete Schrödinger-type operators in d -dimensional lattice Z d , where ∆ is the discrete Laplacian and v is of rank-one. We prove that there exist coupling constant thresholds µo, µ o ≥ 0 such that for any µ ∈ [−µ o , µo] the discrete spectrum of hµ is empty and for any µ ∈ R \ [−µ o , µo] the discrete spectrum of hµ is a singleton {e(µ)}, and e(µ) < 0 for µ > µo and e(µ) > 4d 2 for µ < −µ o . Moreover, we study the asymptotics of e(µ) as µ → µo and µ → −µ o as well as µ → ±∞. The asymptotics highly depend on d and v.

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Solution of stokes flow problem using biharmonic equation formulation and multiquadrics method

Cited by 7 publications

References 9 publications

Expansion of eigenvalues of the perturbed discrete bilaplacian

Expansion of eigenvalues of the perturbed discrete bilaplacian

Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian

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