Fourth‐order schemes for the initial boundary value problem of the 3‐D acoustic equation

1989 ◽  
Author(s):  
Gary Cohen ◽  
Tuong Ha Duong
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Acoustic Equation ◽  
Initial Boundary Value

scholarly journals Integro-differential systems with variable exponents of nonlinearity

2017 ◽  
Vol 15 (1) ◽  
pp. 859-883 ◽  
Author(s):  
Oleh Buhrii ◽  
Nataliya Buhrii
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Existence Theorem ◽  
Fourth Order ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Exponents ◽  
Differential Systems ◽  
Initial Boundary Value

Abstract Some nonlinear integro-differential equations of fourth order with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.

scholarly journals Stability estimation of the generalized solution to the direct problem for the acoustic equation

2021 ◽  
Vol 2092 (1) ◽  
pp. 012005
Author(s):  
Sergey Kabanikhin ◽  
Altyn Nurseitova ◽  
Syrym Kasenov
Keyword(s):  
Boundary Value Problem ◽  
Direct Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Timelike Surface ◽  
Acoustic Equation ◽  
Ill Posed ◽  
Initial Boundary Value

Abstract The initial-boundary value problem for the acoustic equation with data on a timelike surface is considered in this paper. Such a problem arises, for example, if it is required to determine the acoustic pressure inside the region from a fixed response to part of the boundary from the source involved at the same boundary. It is assumed that the medium is at rest up to a certain instant of time and the parameters of the medium, for example, acoustic density, are known. The problem is considered in a triangular domain. The advisability of this was shown in the second half of the last century in the works of Romanov V.G. (for example, [1]), where it was proved that the solution to the direct problem of acoustic is representable as the sum of a singular and a continuous terms. The author has written out the form of the singular part, investigated the problem in an integral statement, and also proved conditional well-posedness theorems for three cases: for a small parameter of the domain, for small data, and for the source representability of the sought solution. It is known that the initial-boundary value problem for the acoustic equation with data on a timelike surface is ill-posed. In this paper, the original ill-posed problem is reduced to an inverse problem with respect to some direct (well-posed) problem. The theorem is proved and a stability estimate of the generalized solution to the direct problem is obtained.

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