title: Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations
metadata.name: dc.contributor.author
metadata.value: Gishlarkaev, Vakha I.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2023-03-14T19:14:46Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2023-03-14T19:14:46Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2023-03
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Gishlarkaev V. I. Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2023, vol. 10 (68), issue 1, pp. 21–35. https://doi.org/10.21638/spbu01.2023.103 (In Russian)
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2023.103
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/39323
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The article proposes a method for analyzing the Cauchy problem for a wide class of evolutionary linear partial differential equations with variable coefficients. By applying the (inverse) Fourier transform, the original equation is reduced to an integro-differential equation, which can be considered as an ordinary differential equation in the corresponding Banach space. The selection of this space is carried out in such a way that the principle of contraction mappings can be used. To carry out the corresponding estimates for the operators generated by the transformed equation, we impose the conditions of finiteness in the space variable for the inverse Fourier transform of the coefficients, and the spaces of the coefficients of the original equation are determined from the Paley-Wiener Fourier transform theorems. In this case, the apparatus of the theory of the Bochner integral in pseudo-normed spaces, countably-normed spaces and Sobolev spaces is used. Classes of functions are distinguished in which the existence and uniqueness of solutions are proved. For equations with coefficients with separated variables, exact solutions are obtained in the form of a Fourier transform of finite sums for operator exponentials.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 10 (68); Issue 1
metadata.lang: -
metadata.name: dc.subject
metadata.value: Fourier transform
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: distributions with compact support
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: convolution
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: integrodifferential equations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Paley-Wiener theorem
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: operator exponential
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 15491-Текст статьи-51842-1-2-20230304.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/39323/1/15491-%d0%a2%d0%b5%d0%ba%d1%81%d1%82%20%d1%81%d1%82%d0%b0%d1%82%d1%8c%d0%b8-51842-1-2-20230304.pdf
resources.format: Adobe PDF
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