Numerical Methods for Hyperbolic and Parabolic Integro-Differential Equations

Pani, Amiya K.; Thomée, Vidar; Wahlbin, Lars B.

doi:10.1216/jiea/1181075713

Cited by 78 publications

(50 citation statements)

References 8 publications

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“…(i) purely numerical methods (classical and modified hybrid variants) such as explicit and implicit (backward) Euler schemes, a discontinuous Galerkin method, energy methods [12], general, standard and Galerkin finite element methods for PIDE, the Tau method [13], the one-step Runge-Kutta and multi-step methods [14,15,16], the Runge-Kutta method [17] for calculation of covariance functions, extrapolation methods [18], Galerkin methods [19], the method of iterations at the last step, a usage of wavelets [20], globally defined Sinc basis functions [21], an approximate transformation of SOIDE into SODE on the base of replacements of kernels with respect to second arguments by piecewise constant functions [22,7], and gamma-distribution expansions;…”

Section: Subject Area Models Review Of Tools and Structure Of The mentioning

confidence: 99%

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

Полосков

Soize

2018

Applied Mathematical Modelling

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To cite this version:Igor E. Poloskov, Christian Soize. Symbolic and numeric scheme for solution of linear integrodifferential equations with random parameter uncertainties and Gaussian stochastic process input. Applied Mathematical Modelling, Elsevier, 2018, 56, pp.15-31 AbstractThe paper describes a theoretical apparatus and an algorithmic part of application of the Green matrix-valued functions for time-domain analysis of systems of linear stochastic integro-differential equations. It is suggested that these systems are subjected to Gaussian nonstationary stochastic noises in the presence of model parameter uncertainties that are described in the framework of the probability theory. If the uncertain model parameter is fixed to a given value, then a time-history of the system will be fully represented by a second-order Gaussian vector stochastic process whose properties are completely defined by its conditional vector-valued mean function and matrix-valued covariance function. The scheme that is proposed is constituted of a combination of two subschemes. The first one explicitly defines closed relations for symbolic and numeric computations of the conditional mean and covariance functions, and the second one calculates unconditional characteristics by the Monte Carlo method. A full scheme realized on the base of Wolfram Mathematica and Intel Fortran software programs, is demonstrated by an example devoted to an estimation of a nonstationary stochastic response of a mechanical system with a thermoviscoelastic component. Results obtained by using the proposed scheme are compared with a reference solution constructed by using a direct Monte Carlo simulation.

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Section: Subject Area Models Review Of Tools and Structure Of The mentioning

confidence: 99%

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

Полосков

Soize

2018

Applied Mathematical Modelling

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“…[11], [12], [13], [14]. Future prospects will be the extension on non-linear problems as the population model in [2], p.868 and on problems with so called weak kernels |K(t)| < Ct −α with 0 < α < 1 for t > 0, see e.g.…”

Section: Conclusion and Future Prospectsmentioning

confidence: 99%

An Adaptive Higher Order Method in Time for Partial Integro‐Differential Equations

Frochte¹

2008

AIP Conference Proceedings

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Abstract. The numerical solution of time-dependent partial integro-differential equations is considered. This includes both variants, the initial value problem and the prehistory problem. One problem concerning partial integro-differential equations is the data storage. To deal with this problem an adaptive method of third order in time is developed to save storage data at smooth parts of the solution. Beyond this a post-processing step adaptively thins out the history data.

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“…Over the last decade, various numerical methods based on finite element approximations in space and special quadrature in time have been developed and studied for this type of problem [20,24,25,29,31,32,34]. The crucial tools used in the analysis are the Ritz and RitzVolterra projections which are instrumental in deriving optimal-order error estimates in various Sobolev norms [5,6,20].…”

Section: Au = −∇·(A∇u) and B(t S)u(s) = −∇·(B∇u(s))mentioning

confidence: 99%

Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data

Sinha¹,

Ewing²,

Lazarov³

2006

SIAM J. Numer. Anal.

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Abstract.A semidiscrete finite volume element (FVE) approximation to a parabolic integrodifferential equation (PIDE) is analyzed in a two-dimensional convex polygonal domain. An optimalorder L 2 -error estimate for smooth initial data and nearly the same optimal-order L 2 -error estimate for nonsmooth initial data are obtained. More precisely, for homogeneous equations, an elementary energy technique and a duality argument are used to derive an error estimate of order O t −1 h 2 ln h in the L 2 -norm for positive time when the given initial function is only in L 2 .

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Numerical Methods for Hyperbolic and Parabolic Integro-Differential Equations

Cited by 78 publications

References 8 publications

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

An Adaptive Higher Order Method in Time for Partial Integro‐Differential Equations

Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data

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