Abstract: In this paper, we investigate the solution of a two dimensional parabolic free boundary problem. The free boundary of this problem is modelled as a nonlinear integral equation (IE). For this integral equation, we propose an asymptotic solution as time is near to maturity and develop an integral iterative method. The computational results reveal that our asymptotic solution is very close to the numerical solution as time is near to maturity.
Abstract: In this paper, the American exchange option (AEO) valuation problem is modelled as a free boundary problem. The critical stock price for an AEO is satisfied an integral equation implicitly. When the remaining time is large enough, an asymptotic formula is provided for pricing an AEO. The numerical results reveal that our asymptotic pricing formula is robust and accurate for the long-term AEO.
Abstract: In this paper, the two-dimensional reversed stagnationpoint
flow is solved by means of an anlytic approach. There are
similarity solutions in case the similarity equation and the boundary
condition are modified. Finite analytic method are applied to obtain
the similarity velocity function.
Abstract: This paper study the behavior of the solution at the crack edges for an elliptical crack with developing cusps, Ω in the plane elasticity subjected to shear loading. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over Ω and it is then transformed into a similar equation over a circular region, D, using conformal mapping. An appropriate collocation points are chosen on the region D to reduce the hypersingular integral equation into a system of linear equations with (2N+1)(N+1) unknown coefficients, which will later be used in the determination of shear stress intensity factors and maximum shear stress intensity. Numerical solution for the considered problem are compared with the existing asymptotic solution, and displayed graphically. Our results give a very good agreement to the existing asymptotic solutions.
Abstract: In this work, we analyze the deformation of surface
waves in shallow flows conditions, propagating in a channel of
slowly varying cross-section. Based on a singular perturbation
technique, the main purpose is to predict the motion of waves by
using a dimensionless formulation of the governing equations,
considering that the longitudinal variation of the transversal section
obey a power-law distribution. We show that the spatial distribution
of the waves in the varying cross-section is a function of a kinematic
parameter,κ , and two geometrical parameters εh
and w ε . The above
spatial behavior of the surface elevation is modeled by an ordinary
differential equation. The use of single formulas to model the varying
cross sections or transitions considered in this work can be a useful
approximation to natural or artificial geometrical configurations.