Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity

Saedpanah, Fardin

doi:10.1016/j.euromechsol.2013.10.014

Cited by 23 publications

(37 citation statements)

References 34 publications

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“…Therefore, here we extend [1] to local/global Galerkin approximation methods, based on the Picard's iteration, that gives a straightforward and constructive proof and the only property of the kernel to be used is integrability.…”

Section: K(t − S)au(s) Ds = F (T) T ∈ (0 T ) With U(0)mentioning

confidence: 99%

“…(throughout we useu or u (1) for du dt is considered to be either smooth (exponential), or no worse than weakly singular, that is singular at the origin but locally integrable. We assume that the kernel has the properties…”

Section: K(t − S)au(s) Ds = F (T) T ∈ (0 T ) With U(0)mentioning

confidence: 99%

“…We study, for any fixed T > 0, integro-differential equations of the form u(t) + Au(t) − t 0 K(t − s)Au(s) ds = f (t), t ∈ (0, T ), with u(0) = u 0 ,u(0) = u 1 , (1.1) (throughout we useu or u (1) for du dt ) together with a homogeneous Dirichlet boundary condition on a bounded polygonal domain ⊂ R d , d = 1, 2, 3 with boundary ∂ , or with mixed homogeneous Dirichlet and nonhomogeneous Neumann boundary condition, that is important from practical view point. Here, u and f are the solution and the load term, respectively, for example, the displacement vector and the volume load in viscoelasticity.…”

Section: Introductionmentioning

confidence: 99%

“…( 1.2) Examples of this type of problems, that appear for example, in the theory of linear and fractional order viscoelasticity, is found in Ref. [1], and references therein. The Mittag-Leffler type kernels, as a chief example of fractional order kernels, is used in fractional order viscoelasticity, and interpolates between weakly singular kernels and smooth (exponential) kernels, that arise in the theory of linear viscoelasticity.…”

Section: Introductionmentioning

confidence: 99%

“…There is an extensive literature on theoretical and numerical analysis for partial differential equations (PDEs) with memory, in particular integro-differential equations. To mention some, see [1][2][3][4][5][6][7][8][9][10], and their references.…”

Section: Introductionmentioning

confidence: 99%

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Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

Saedpanah

2015

Numerical Methods Partial

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We study a second order hyperbolic initial-boundary value partial differential equation (PDE) with memory that results in an integro-differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example.

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Section: K(t − S)au(s) Ds = F (T) T ∈ (0 T ) With U(0)mentioning

confidence: 99%

Section: K(t − S)au(s) Ds = F (T) T ∈ (0 T ) With U(0)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

Saedpanah

2015

Numerical Methods Partial

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A global solvability of a two‐dimensional kernel determination problem for a viscoelasticity equation

Totieva

2022

Math Methods in App Sciences

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Under study is the two‐dimensional inverse problem of determining the convolutional kernel of the integral term in a viscoelasticity equation. The direct problem is represented by a generalized initial‐boundary value problem for this equation with zero initial data and the Neumann boundary condition in the form of the Dirac delta function. For solving the inverse problem, the traces of the solution to the direct problem on the domain boundary are given as an additional condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions twice continuously differentiable in the time variable t$$ t $$ and analytic in the space variable. We apply the methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions.

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An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media

Huang

Wang

2018

Adv Comput Math

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In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for Maxwell equations in Cole-Cole dispersive media in twodimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields and the induced electric polarisation (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then we solve to IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show such an enriched collocation method is convergent and accurate.2010 Mathematics Subject Classification. 65N35, 65E05, 65N12, 41A10, 41A25, 41A30, 41A58.

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Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity

Cited by 23 publications

References 34 publications

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

A global solvability of a two‐dimensional kernel determination problem for a viscoelasticity equation

An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media

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