Finite element methods for parabolic and hyperbolic partial integro-differential equations

Yanik, Elizabeth Greenwell; Fairweather, Graeme

doi:10.1016/0362-546x(88)90039-9

Cited by 159 publications

(83 citation statements)

References 19 publications

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“…Then siny'0 > 0 for j < 8, and since | sin jd\ < j\ sin8\, we have by (A.l), The interval I2 = (it¡3, ic/2) will be further divided into I'2 = (n/3, 2n/5) and V{ -(2n/5, it/2). The following table shows upper bounds for Jf (d) and J29 (8) in the respective interval, which will be shown below.…”

Section: Jomentioning

confidence: 99%

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Long-time numerical solution of a parabolic equation with memory

Thom{ée¹,

Wahlbin²

1994

Math. Comp.

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Abstract. Long-time stability and convergence properties of two time-discretization methods for an integro-differential equation of parabolic type are studied. The methods are based on the standard backward Euler and second-order backward differencing methods. The memory term is approximated by a quadrature rule, with emphasis on such rules with reduced computational memory requirements. Discretization of the spatial partial differential operators by the finite element method is also considered.

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Section: Jomentioning

confidence: 99%

“…We refer to [2, §3] in particular, for further details. Other examples are briefly described in [8], and some further examples referenced in [6].…”

Section: Introductionmentioning

confidence: 99%

Long-time numerical solution of a parabolic equation with memory

Thom{ée¹,

Wahlbin²

1994

Math. Comp.

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“…Parabolic Volterra integro-differential equations have many important physical applications to model dynamical systems, such as in compression of viscoelastic media [4], nuclear reactor dynamics [5], blow-up problems [6], reaction diffusion problems [7] and heat conduction materials with memory functional [8], etc. So far, the analysis of numerical solution of Volterra integral-differential equations has been carried out by many authors.…”

Section: Introductionmentioning

confidence: 99%

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

Zhao

2016

Entropy

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This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of the Fox-H function and convolution theorem, analytical solutions for the equations in the infinite domain are derived under three frictional memory kernel functions. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, the Fox-H function and the convolution form of the Fourier transform. In addition, graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equations with a power-law memory kernel. It can be seen that the solution curves are subject to Gaussian decay at any given moment.

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“…For equations with nonsmoofh kernels such as in (1.2), we refer to Grimmer and Pritchard [4], Lunardi and Sinestrari [10], and Lorenzi and Sinestrari [9] and references therein. Finite element methods for problems of the form (1.1) with a smooth kernel K have been discussed in, e.g., Sloan and Thomée [13], Yanik and Fairweather [15], Thomée and Zhang [14], LeRoux and Thomée [6], Cannon and Lin [1], and Lin, Thomée, and Wahlbin [7].…”

Section: Introductionmentioning

confidence: 99%