An elastic half-space under the action of one-dimensional time-dependent diffusion perturbations

Thermoelastic Diffusion Multicomponent Half-Space under the Effect of Surface and Bulk Unsteady Perturbations

“…We apply the Laplace transform on time to the homogeneous system of Equation 10and the boundary conditions Equation 11, and get a system of differential equations [17][18][19][20] («s» is the transformation parameter, the index «L» denotes the Laplace transformant):…”

“…Then, we integrate it along Ox in the interval from 0 to +∞ and take into account Equation (13). As a result, we obtain the system of linear algebraic equations for the Fourier-Laplace images [17,20]:…”

“…The originals in Equation (19) can be calculated using well-known numerical algorithms, such as inverse DFT-type methods or the Filon's method. The quadrature formula on an irregular sparsening grid presented in [17] can also be used.…”

“…The solution of the problem, similar to problems discussed in [15][16][17][18][19][20], is sought in the integral convolution form of Green's functions and functions of bulk or surface external perturbations. The Laplace transform on time and the Fourier sine and cosine transform on the coordinate are used to find the Green's functions.…”

“…Only the roots s j of P(λ, s) and the inverse sine and cosine transforms were calculated numerically. For inverse transforms we used the quadrature formula on an irregular sparsening grid [17], with 10 5 integration points. This method is quite fast, but it has a slow convergence over the grid splitting.…”