A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

Radu, Petronela; Toundykov, Daniel; Trageser, Jeremy

doi:10.1007/s00205-016-1047-2

Cited by 19 publications

(14 citation statements)

References 20 publications

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“…A direct consequence of the characterization of the multipliers in Theorem 1 and the continuity of 2 F 3 is given by the following result. Convergence of peridynamic operators (in both scalar and vector cases) to the corresponding differential operators in the limit of vanishing nonlocality δ → 0 + is well-know, see for example [3,11,14,18,19]. Corollary 1 recovers the convergence L δ,β → ∆, as δ → 0 + , but in the sense of convergence of the multipliers.…”

Section: Convergence To the Laplacianmentioning

confidence: 84%

“…For scalar-valued kernels γ and scalar fields u, the operators in (1) have been studied in the context of nonlocal diffusion, digital image correlation, and nonlocal wave phenomena among other applications, see for example [5,7,16,20]. Several mathematical and numerical studies have focused on nonlocal Laplace operators including [2,10,19]. In this work, we focus on radially symmetric kernels with compact support of the form…”

Section: Introductionmentioning

confidence: 99%

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Fourier multipliers for nonlocal Laplace operators

Alali

Albin

2019

Applicable Analysis

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Fourier multiplier analysis is developed for peridynamic Laplace operators, which are defined for scalar fields in R n . The focus is on operators L δ,β with compactly supported integral kernels of the form χ B δ (0) (x) 1x β , which are commonly used in peridynamic models [6]. The Fourier multipliers m(ν) of L δ,β are given through an integral representation, which is shown to be well-defined for β < n + 2. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function 2 F 3 in any spatial dimension n and for any β < n + 2. The new general formula for the multipliers is well-defined for any β ∈ R \ {n + 4, n + 6, n + 8, . . .} and is utilized to extend the definition of the peridynamic Laplacians to the case when β ≥ n + 2 (with β = n + 4, n + 6, n + 8, . . .). Some special cases are presented. We show that the Fourier multipliers of L δ,β converge to the Fourier multipliers of ∆ in the limit as β → n + 2. In addition, we show that in the limit as β → n + 2 − , L δ,β u converges to ∆u for sufficiently regular u. Moreover, we identify the limit of the Fourier multipliers of L δ,β as β → −∞ and, furthermore, recognize the limit operator L δ,−∞ as an integral operator with kernel supported in the (n − 1)-sphere.Asymptotic analysis of 2 F 3 is utilized to identify the asymptotic behavior of the Fourier multipliers as ν → ∞. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel (i.e., when β < n), diverge to −∞ at a rate proportional to − log( ν ) when β = n, and diverge to −∞ at a rate proportional to − ν β−n when β > n. The bounds and decay rates are presented explicitly in terms of n, β, and the nonlocality δ.In periodic domains, the Fourier multipliers are the eigenvalues of L δ,β and therefore the presented results provide characterization formulas and asymptotic behavior for the eigenvalues. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation.

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Section: Convergence To the Laplacianmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Fourier multipliers for nonlocal Laplace operators

Alali

Albin

2019

Applicable Analysis

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“…The operators have an integral form which collect information in a neighborhood of a point through a kernel of interaction. Nonlocal interactions in a variety of applications have been expressed through different operators (single, or double convolution-type operators), for which results connecting the local and nonlocal frameworks have been established [2,4,10,14,21,22,24,25].…”

Section: Introductionmentioning

confidence: 99%

Sensitivity analysis for solutions to heterogeneous nonlocal systems

Buczkowski,

Foss,

Parks

et al. 2021

Preprint

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The paper presents a collection of results on continuous dependence for solutions to nonlocal problems under perturbations of data and system parameters. The integral operators appearing in the systems capture interactions via heterogeneous kernels that exhibit different types of weak singularities, space dependence, even regions of zero-interaction. The stability results showcase explicit bounds involving the measure of the domain and of the interaction collar size, nonlocal Poincaré constant, and other parameters. In the nonlinear setting the bounds quantify in different L p norms the sensitivity of solutions under different nonlinearity profiles. The results are validated by numerical simulations showcasing discontinuous solutions, varying horizons of interactions, and symmetric and heterogeneous kernels.N.B, M.F, P.R. were supported by the award NSF -DMS 1716790

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“…Peridynamic operators of the form (1) have been used in different applications including nonlocal diffusion, digital image correlation, and nonlocal wave phenomena, see for example [4,6,[15][16][17]19]. Nonlocal equations that involve Laplace-type operators have been addressed in several mathematical and numerical studies including [1,2,8,18]. These nonlocal Laplace operators are motivated by the peridynamic theory for continuum mechanics [5,20,21] and were first introduced in nonlocal vector calculus [9].…”

Section: Introductionmentioning

confidence: 99%

Fourier Spectral Methods for Nonlocal Models

Alali

Albin

2020

J Peridyn Nonlocal Model

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Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in Alali and Albin (Appl Anal :1-21, 2019). It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in n dimensions reduces to the computation of a 1D smooth function given in terms of 2 F 3. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are discussed.

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A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

Cited by 19 publications

References 20 publications

Fourier multipliers for nonlocal Laplace operators

Fourier multipliers for nonlocal Laplace operators

Sensitivity analysis for solutions to heterogeneous nonlocal systems

Fourier Spectral Methods for Nonlocal Models

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