title: Fourier transform method for partial differential equations: Formulas for representing solutions to the Cauchy problem
metadata.name: dc.contributor.author
metadata.value: Gishlarkaev, Vakha I.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2022-10-20T16:32:11Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2022-10-20T16:32:11Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2022-09
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Gishlarkaev V. I. Fourier transform method for partial differential equations: Formulas for representing solutions to the Cauchy problem. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2022, vol. 9 (67), issue 3, pp. 480–494. https://doi.org/10.21638/spbu01.2022.309 (In Russian)
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2022.309
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/38306
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The paper proposes a method for solving the Cauchy problem for linear partial differential
  equations with variable coefficients of a special form, allowing, after applying the (inverse)
  Fourier transform, to rewrite the original problem as a Cauchy problem for first-order partial
  differential equations. The resulting problem is solved by the method of characteristics
  and the (direct) Fourier transform is applied to its solution. And for this it is necessary to
  know the solution of the Cauchy problem for a first-order equation in the entire domain of
  definition. This leads to the requirement that the support of the (inverse) Fourier transform
  of the initial function of the original problem be compact, and to describe the class of
  initial functions, it is necessary to use Paley—Wiener—Schwarz-type theorems on Fourierimages,
  including distributions. The presentation of solutions in the form of the Fourier
  transform of some function (distribution), determined by the initial function, is presented.
  A general form of the evolutionary equation is written down, which, when the described
  method is applied, leads to the consideration of a homogeneous first-order equation, and a
  formula for the solution of the Cauchy problem in this general case is derived. The general
  form of the equation is written down, which leads to the consideration of a first-order inhomogeneous
  equation, and a formula for solutions it is derived. Particular cases of these
  equations are the well-known equations that are encountered in the description of various
  processes in physics, chemistry, and biology.
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 9 (67); Issue 3
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: method of characteristics
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Fourier transform method for partial differential equations: Formulas for representing solutions to the Cauchy problem
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 480-494.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/38306/1/480-494.pdf
resources.format: Adobe PDF
resources.size: 352,04 kB