This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

We obtain all the solutions of types u(x, y) = f(x) + g(y) and u(x, y) = f(x)g(y) for three known mean-curvature-prescribed equations, namely, the capillary equation, the translating soliton equation and the two-dimensional analogue of the catenary.

We use a method based on a separation of variables for solving a first order partial differential equations system, using a very simple modelling of MHD. The method consists in introducing three unknown variables Φ 1, Φ 2, Φ 3 in addition to the time variable t and then in searching a solution which is separated with respect to Φ 1 and t only. This is allowed by a very simple relation, called a “metric separation equation”, which governs the type of solutions with respect to time. The families of solutions for the system of equations thus obtained, correspond to a radial evolution of the fluid. Solving the MHD equations is then reduced to find the transverse component H ∑ of the magnetic field on the unit sphere Σ by solving a non linear partial equation on Σ. Thus, we generalize ideas of Courant-Friedrichs [7] and of Sedov [11], on dimensional analysis and self-similar solutions.

We construct an elliptic analogue of Sklyanin‘s separation of variables for the sl(2) Gaudin system, using an adaptation of Drinfeld‘s Radon transformations.

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.