Generalized integral representation method (GIRM) is designed to numerically solve initial and boundary value problems for differential equations. In this work, we develop numerical schemes based on 1- and 2-step GIRMs for evaluation of the two-dimensional problem of advective diffusion in an infinite domain. Accurate approximate solutions are obtained in both cases of GIRM and compared to the exact ones. The derivation of GIRM is straightforward and implementation is simple.
Numerical evaluations of soliton-soliton and soliton-to-bottom interaction have many applications in various fields. On the other hand, Generalized Integral Representation Method (GIRM) is known as a convenient numerical method for solving Initial and Boundary Value Problem of differential equations such as advective diffusion. In this work, we apply one-step GIRM to numerical evaluations of propagation of a single soliton, soliton-to-soliton interaction and soliton-to-bottom interaction. Firstly, in case of a single soliton, the bottom is considered to be constant in order to understand the behavior of the soliton propagation as it travels in the middle of the sea. Next, in case of soliton-to-bottom, we study behavior of a single soliton propagation when the bottom has different geometries. Finally, we evaluate interaction of two different i.e., big and small solitons. To carry out with the studies, we derive and implement GIRM to numerically solve the Korteweg-de Vries (KdV) equation. In order to verify the theory, numerical experiments are conducted and accurate approximate solutions are obtained in each case of the soliton interactions.
In this paper, we are proposing a novel method to estimate static displacements of atoms caused by size effects in fcc substitutional binary polycrystalline solid solutions. Fourier transforms of static displacements of the atoms on every considered shell were calculated using the equations that include dynamical matrix and Fourier transform of interatomic forces. Short-range order parameters on the first seven shells of Ni-14 at. % Ir alloy have been identified from X-ray diffuse scattering intensity by accounting microscopic static displacements of atoms on a particular shell. Pairwise interatomic potentials on the considered shells and critical temperature of disorder-order phase transition were calculated using values of short-range order parameters.
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