Open AccessOn the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation
Author(s) -
Pavel N. Burago,
Albert Egamov
Publication year - 2019
Publication title -
žurnal srednevolžskogo matematičeskogo obŝestva
Language(s) - English
Resource type - Journals
eISSN - 2587-7496
pISSN - 2079-6900
DOI - 10.15507/2079-6900.21.201904.413-429
Subject(s) - mathematics , boundary value problem , mathematical analysis , hyperbolic partial differential equation , connection (principal bundle) , operator (biology) , fourier integral operator , initial value problem , partial differential equation , nonlinear system , differential equation , integral equation , geometry , biochemistry , chemistry , physics , repressor , quantum mechanics , transcription factor , gene
We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.