Mathematical Formulation of Inverse Scattering and Korteweg-De Vries Equation

Inverse scattering refers to the determination of the solutions of a set of differential equations based on known asymptotic solutions, that is, the solution of Marchenko equation. Marchenko equation was derived using integral equation. The potential function derived from eigenvalues and scattering data seems to be the inverse method of scattering problem. The reflection coefficient with one pole and zero reflection coefficients has been chosen to solve inverse scattering problem. Again this paper deals with the connection between inverse scattering and the Korteweg-de Vries equation and describes variety of examples with Korteweg-de Vries equation: the single-soliton solution, the two-soliton solution and finally the N-soliton solution. Throughout the work, the primary objective is to study some mathematical techniques applied in analyzing the behavior of soliton in the KdV equations. Keywords: Marchenko equation, KdV equation, Solitons, Scattering, Inverse Scattering, Canal

Transient thermophoretic particle deposition on forced convective heat and mass transfer flow due to a rotating disk

This paper investigates thermophoretic deposition of micron sized particles on unsteady forced convective heat and mass transfer flow due to a rotating disk. Using similarity transformations the governing nonlinear partial differential equations are transformed into a system of ordinary differential equations that are then solved numerically by applying Nachtsheim–Swigert shooting iteration technique along with sixth-order Runge–Kutta integration scheme. The effects of the pertinent parameters on the radial, tangential and axial velocities, temperature and concentration distributions, and axial thermophoretic velocity together with the local skin-friction coefficient, and local Nusselt number are displayed graphically. The inward axial thermophoretic deposition velocity (local Stanton number) is also tabulated. The obtained results show that axial thermophoretic velocity is increased with the increasing values of the thermophoretic coefficient, thermophoresis parameter, rotational parameter as well as unsteadiness parameter. The results also show that inward axial thermophoretic particle deposition velocity decreases with the increase of the Lewis number