Morphological Evolution in Heteroepitaxial Thin Film Structures at the Nanoscale

The aim of this study is to resolve the phenomenon of formation of mesoscopic structures on the surface of heteroepitaxial thin film system due to surface diffusion by considering the effects of both surface and interface stresses. Elastic stress field caused by curved surface is solved by using the constitutive equations of linear elasticity for the bulk and surface phases. Based on the method of superposition, a boundary perturbation technique, Goursat-Kolosov complex potentials and Muskhelishvili representations, the boundary value problem is reduced to the successive solution of a system of singular and hypersingular integral equations for any order of approximation. This solution and thermodynamic approach allows us to derive a governing equation which gives the amplitude changing of a surface roughness with time.

Solving a Class of Singular Two-Point Boundary Value Problems Using New Modified Decomposition Method

We introduce an effective methodology for solving a class of linear as well as nonlinear singular two-point boundary value problems. This methodology is based on a modification of Adomian decomposition method (ADM) and a new two fold integral operator. We use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components of solution. Thus, we develop modified recursive scheme without any undetermined coefficients while computing the successive solution components. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. However, most of earlier recursive schemes using ADM do require computation of undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the present technique. The results reveal that the method is very effective, straightforward, and simple.

New Approach for Solving a Class of Doubly Singular Two-Point Boundary Value Problems Using Adomian Decomposition Method

We propose two new modified recursive schemes for solving a class of doubly singular two-point boundary value problems. These schemes are based on Adomian decomposition method (ADM) and new proposed integral operators. We use all the boundary conditions to derive an integral equation before establishing the recursive schemes for the solution components. Thus we develop recursive schemes without any undetermined coefficients while computing successive solution components, whereas several previous recursive schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete calculation of the solution by several earlier modified recursion schemes using the ADM. The approximate solution is computed in the form of series with easily calculable components. The effectiveness of the proposed approach is tested by considering four examples and results are compared with previous known results.