Sentence examples for boundary integrals can be from inspiring English sources
The nearly singular boundary integrals can be evaluated accurately with the previously developed distance transformation techniques.
The approach is based on the fact that the singular boundary integrals can be represented approximately by the mean values of two nearly singular boundary integrals and on the techniques of distance transformations developed primarily in previous work of the authors.
This is because any CPV boundary integral can be approximated by the mean value of two corresponding nearly singular boundary integrals at boundary points where the continuity requirements are met.
By using the relation z * conjg z) = a * a, or conjg z) = a * a/z on the circular boundary with radius a, all integrals can be reduced to an integral for complex variable and they can be integrated in closed form.
The other integral can be evaluated numerically.
The integral can be approximated with Gauss-Hermite quadratures.
Even when the partial differential equation has a unique solution, for any given closed boundary I, these elementary boundary integral equations can be shown to be singular at a countable set of characteristic wavenumbers.
The boundary integral equations can be simplified by using the Green's functions as the kernels.
By using the one-dimensional wave propagation mode, the degree of the singularities appearing in the conventional boundary integral equation can be reduced.
The boundary integral equations can be written in two equivalent forms: (a) the tractions can be written as a space time convolution of the displacement continuities at the interface (Budiansky and Rice, 1979) (b) the displacement discontinuities can be written as a space time convolution of the tractions at the interface (Kostrov, 1966).
For our simple model problem Γ = Γ 0. Note that we distinguish Γ 0 and Γ from the beginning, such that the developed boundary integral equations can be applied to more general settings which are discussed in Section 6.
