412 research outputs found
On the local transformed based method for partial integro-differential equations of fractional order
Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity
An integro-differential equation, modeling dynamic fractional order
viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A
discontinuous Galerkin method, based on piecewise constant polynomials is
formulated for temporal semidiscretization of the problem. Stability estimates
of the discrete problem are proved, that are used to prove optimal order a
priori error estimates. The theory is illustrated by a numerical example.Comment: 16 pages, 2 figure
F. John's stability conditions vs. A. Carasso's SECB constraint for backward parabolic problems
In order to solve backward parabolic problems F. John [{\it Comm. Pure. Appl.
Math.} (1960)] introduced the two constraints "" and where satisfies the backward heat equation for
with the initial data
The {\it slow-evolution-from-the-continuation-boundary} (SECB) constraint has
been introduced by A. Carasso in [{\it SIAM J. Numer. Anal.} (1994)] to attain
continuous dependence on data for backward parabolic problems even at the
continuation boundary . The additional "SECB constraint" guarantees a
significant improvement in stability up to In this paper we prove that
the same type of stability can be obtained by using only two constraints among
the three. More precisely, we show that the a priori boundedness condition
is redundant. This implies that the Carasso's SECB condition
can be used to replace the a priori boundedness condition of F. John with an
improved stability estimate. Also a new class of regularized solutions is
introduced for backward parabolic problems with an SECB constraint. The new
regularized solutions are optimally stable and we also provide a constructive
scheme to compute. Finally numerical examples are provided.Comment: 15 pages. To appear in Inverse Problem
An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an -version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal -version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the -version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems
A spectral order method for inverting sectorial Laplace transforms
Laplace transforms which admit a holomorphic extension to some sector
strictly containing the right half plane and exhibiting a potential behavior
are considered. A spectral order, parallelizable method for their numerical
inversion is proposed. The method takes into account the available information
about the errors arising in the evaluations. Several numerical illustrations
are provided.Comment: 17 pages 11 figure
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