Geodesic
On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 413-429

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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.
Keywords: the second initial boundary value problem, integro-differential equation with PDE, phase constraint, hyperbolic equation. 
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     author = {P. N. Burago and A. I. Egamov},
     title = {On the connection between solutions of initial boundary-value problems for a some class of integro-differential {PDE} and a linear hyperbolic equation},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {413--429},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a1/}
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P. N. Burago; A. I. Egamov. On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 4, pp. 413-429. http://geodesic.mathdoc.fr/item/SVMO_2019_21_4_a1/