Geodesic
Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem
Serdica Mathematical Journal, Tome 41 (2015) no. 4, pp. 415-430.

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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.
Keywords: convolutional calculus, non-classical convolution, Duhamel principle, ill-posed problem, quasi-reversibility 
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Bazhlekova, Emilia. Convolutional Calculus of Dimovski and QR-regularization of the Backward Heat Problem. Serdica Mathematical Journal, Tome 41 (2015) no. 4, pp. 415-430. http://geodesic.mathdoc.fr/item/SMJ2_2015_41_4_a5/