Method of successive approximations in stochastic elastic boundary value problem for structurally heterogenous materials

2012 ◽  
Vol 52 (1) ◽  
pp. 101-106 ◽  
Author(s):  
Mikhail Tashkinov ◽  
Valeriy Wildemann ◽  
Natalia Mikhailova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Elastic Boundary

Interaction of acoustic waves with moving boundary surfaces

2021 ◽  
pp. 59-60
Author(s):  
Keyword(s):  
Boundary Value Problem ◽  
Acoustic Waves ◽  
Moving Boundary ◽  
Boundary Value ◽  
Boundary Surface ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Ultrasonic Beam ◽  
Static Approximation ◽  
Quasi Static Approximation

A quasi-static approximation is considered for the interaction of a probing ultrasonic beam with a vibrating boundary surface. The model is considered in the form of a boundary value problem, presented in the form of d'Alembert. The method of successive approximations was used for the solution. The error arising from this interaction is established. Keywords: quasi-static approximation, boundary value problem, d'Alembert form, Doppler effect, rheological medium. [email protected]; [email protected]

scholarly journals An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
T. S. Jang
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Successive Approximations ◽  
Initial Boundary Value

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-boundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.

Laplacian structure mirroring surface topography in determining the gravity potential by successive approximations

2021 ◽  
Author(s):  
Petr Holota ◽  
Otakar Nesvadba
Keyword(s):  
Boundary Value Problem ◽  
Laplace Operator ◽  
Boundary Value ◽  
Successive Approximations ◽  
Gravity Potential ◽  
Geodetic Boundary Value Problem ◽  
Physical Surface ◽  
The Earth ◽  
Geodetic Boundary ◽  
The Laplace Operator

<p>Similarly as in other branches of engineering and mathematical physics, a transformation of coordinates is applied in treating the geodetic boundary value problem. It offers a possibility to use an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. In our case the Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth and thus also the solution domain substantially differ from a sphere or an oblate ellipsoid of revolution, even if optimally fitted. The situation becomes more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces. Applying tensor calculus the Laplace operator is expressed in the new coordinates. However, its structure is more complicated in this case and in a sense it represents the topography of the physical surface of the Earth. The Green’s function method together with the method of successive approximations is used for the solution of the geodetic boundary value problem expressed in terms of the new coordinates. The structure of iteration steps is analyzed and if useful and possible, it is modified by means of integration by parts. Subsequently, the iteration steps and their convergence are discussed and interpreted, numerically as well as in terms of functional analysis.</p>

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