Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem

Bazhlekova, Emilia. "Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem." Serdica Mathematical Journal 41.4 (2015): 415-430. <http://eudml.org/doc/295049>.

@article{Bazhlekova2015,
abstract = {[Bazhlekova Emilia; Бажлекова Емилия]The final value problem for the heat equation is known to be ill-posed. To deal with this, in the method of quasi-reversibility (QR), the equation or the final value condition is perturbed to form an approximate well-posed problem, depending on a small parameter ε. In this work, four known quasi-reversibility techniques for the backward heat problem are considered and the corresponding regularizing problems are treated using the convolutional calculus approach developed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht, 1990). For every regularizing problem, applying an appropriate bivariate convolutional calculus, a Duhamel-type representation of the solution is obtained. It is in the form of a convolution product of a special solution of the problem and the given final value function. A non-classical convolution with respect to the space variable is used. Based on the obtained representations, numerical experiments are performed for some test problems. 2010 Mathematics Subject Classification: 35C10, 35R30, 44A35, 44A40.},
author = {Bazhlekova, Emilia},
journal = {Serdica Mathematical Journal},
keywords = {convolutional calculus; non-classical convolution; Duhamel principle; ill-posed problem; quasi-reversibility},
language = {eng},
number = {4},
pages = {415-430},
publisher = {Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences},
title = {Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem},
url = {http://eudml.org/doc/295049},
volume = {41},
year = {2015},
}

TY - JOUR
AU - Bazhlekova, Emilia
TI - Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem
JO - Serdica Mathematical Journal
PY - 2015
PB - Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences
VL - 41
IS - 4
SP - 415
EP - 430
AB - [Bazhlekova Emilia; Бажлекова Емилия]The final value problem for the heat equation is known to be ill-posed. To deal with this, in the method of quasi-reversibility (QR), the equation or the final value condition is perturbed to form an approximate well-posed problem, depending on a small parameter ε. In this work, four known quasi-reversibility techniques for the backward heat problem are considered and the corresponding regularizing problems are treated using the convolutional calculus approach developed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht, 1990). For every regularizing problem, applying an appropriate bivariate convolutional calculus, a Duhamel-type representation of the solution is obtained. It is in the form of a convolution product of a special solution of the problem and the given final value function. A non-classical convolution with respect to the space variable is used. Based on the obtained representations, numerical experiments are performed for some test problems. 2010 Mathematics Subject Classification: 35C10, 35R30, 44A35, 44A40.
LA - eng
KW - convolutional calculus; non-classical convolution; Duhamel principle; ill-posed problem; quasi-reversibility
UR - http://eudml.org/doc/295049
ER -