Dual integral equation method

Our method of attacking plane, non-inertial problems will be, in the first instance, to reduce (2.8.9) to a Hilbert problem, in precisely the manner developed by Muskhelishvili (1963), and then to handle the specifically viscoelastic aspects, essentially by the methods outlined in Sects. 2.4-6. We remark that an alternative way of approaching the first stage is the dual integral equation method originally used in this context by Sneddon (1951) but with a long history of mathematical development summarized by Gladwell (1980). 

In the present paper the fracture is treated as a real crack with infinite small distance between sides. For the solution of the elasticity problem with the cavity and the fracture, the modification of the Dual BEM with discontinuous elements is built [4]. It is the most optimal method with regard to the computational costs and the convenience of the integral equations approximation. Near the crack front special elements are used. They account the singularity of the elasticity problem solution. To improve the accuracy of the Stress Intensity Factors calculation, the special boundary elements near the crack front are accounted in the interpolation formulae. 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. Satisfactory agreement is obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves (time-lag method). These rate curves were also compared to the curves predicted by Fickian-type diffusion equations. 

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