Nonlinear second order systems of Fredholm integro-differential equations

SeMA Journal ◽  
2021 ◽  
Author(s):  
Mohamed El-Gamel ◽  
Ola Mohamed
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Second Order Systems

On the instability in second-order systems for acoustic VTI and TTI media

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. T171-T186 ◽  
Author(s):  
Kenneth P. Bube ◽  
Tamas Nemeth ◽  
Joseph P. Stefani ◽  
Ray Ergas ◽  
Wei Liu ◽  
...  
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Differential Equations ◽  
Shear Wave ◽  
Transversely Isotropic ◽  
Second Order ◽  
Wave Speeds ◽  
Tti Media ◽  
Second Order Systems ◽  
Vti Media

We studied second-order wave propagation systems for vertical transversely isotropic (VTI) and tilted transversely isotropic (TTI) acoustic media with variable axes of symmetry that have their shear-wave speeds set to zero. Acoustic TTI systems are commonly used in reverse-time migration, but these second-order systems are susceptible to instablities appearing as nonphysical stationary noise growing linearly in time, particularly in variable-tilt TTI media. We found an explanation of the cause of this phenomenon. The instabilities are not caused only by the numerical schemes; they are inherent to the differential equations. These instabilities are present even in homogeneous VTI media. These instabilities are caused by zero wave speeds at a wide variety of wavenumbers — a direct consequence of setting the shear-wave speeds to zero — coupled with the second time derivative in these systems. Although the second-order isotropic wave equation allows smooth time-growing solutions, a larger class of time-growing solutions exists for the second-order acoustic TI systems, including nonsmooth solutions. Boundary conditions appear to be less effective in controlling these time-growing solutions than they are for the isotropic wave equation. These systems conserve an incomplete energy that does not prevent the instabilities. The corresponding steady-state systems are no longer elliptic differential equations and can have nonsmooth solutions that are related to the instabilities. We started initially with homogeneous VTI media, and then extended these results to heterogeneous variable-tilt TTI media. We also developed a second-order acoustic system for heterogeneous variable-tilt TTI media derived directly from the full-elastic system for heterogeneous variable-tilt TTI media. All second-order systems with a dispersion relation obtained by setting the shear-wave speeds to zero in the elastic dispersion relation allowed these nonphysical time-growing solutions; however, knowing the cause of these instabilities, it may be possible to prevent or control the activation of these solutions.

Some second-order systems of partial differential equations of composite type

1987 ◽  
Vol 106 (1-2) ◽  
pp. 73-88 ◽  
Author(s):  
Lin Wei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hyperbolic Systems ◽  
Second Order ◽  
Cauchy Type ◽  
Linear Transformations ◽  
Partial Differential ◽  
Composite Systems ◽  
Composite Type ◽  
Second Order Systems

SynopsisThe Cauchy problem and the Dirichlet-Cauchy type problem of some second-order systems of partial differential equations of composite type of two unknown functions are investigated. Such systems possess some of the characteristics not only of elliptic but also of hyperbolic systems in the same domain. Representations of the solutions are found for the upper half plane. To this end, the composite systems are reduced to the canonical form by means of successive applications of three kinds of linear transformations. Function theoretic methods are used to obtain representation formulae. Furthermore, some composite systems of 2m-unknown function are also considered.

scholarly journals Second-Order Systems With Singular Mass Matrix and an Extension of Guyan Reduction

1995 ◽  
Author(s):  
Sanjay P. Bhat ◽  
Dennis S. Bernstein
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Mass Matrix ◽  
Regular System ◽  
Second Order ◽  
Order System ◽  
Coupled Subsystems ◽  
Singular Mass Matrix ◽  
Guyan Reduction ◽  
Second Order Systems

Abstract The set of consistent initial conditions for a second-order system with singular mass matrix is obtained. In general, such a system can be decomposed (i.e., partitioned) into three coupled subsystems of which the first is algebraic, the second is a regular system of first-order differential equations, and the third is a regular system of second-order differential equations. Under specialized conditions, these subsystems are decoupled. This result provides an extension of Guyan reduction to include viscous damping.

On Oscillation of Solutions of Second-Order Systems of Deviated Differential Equations

1996 ◽  
Vol 3 (6) ◽  
pp. 571-582
Author(s):  
N. Partsvania
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Second Order ◽  
System Of Differential Equations ◽  
Carathéodory Conditions ◽  
Second Order Systems ◽  
Oscillation Of Solutions ◽  
Proper Solutions

Abstract Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations where fi : R + × R 2m → R (i = 1, 2) satisfy the local Carathéodory conditions and σi , τi : R + → R (i = 1, . . . , m) are continuous functions such that σi (t) ≤ t for t ∈ R +, .

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