The numerical solution of a model describing a two-dimensional fluidized bed by a Chebyshev super spectral viscosity (SSV) method is considered. The model is in the form of a hyperbolic system of conservation laws with a source term, coupled with an elliptic equation for determining a stream function. The coupled elliptic equation is solved by a finite difference method. The mixed SSV/finite di...

The conformal thin-sandwich (CTS) equations are a set of four of the Einstein equations, which generalize the Laplace-Poisson equation of Newton's theory. We examine numerically solutions of the CTS equations describing perturbed Minkowski space, and find only one solution. However, we find two distinct solutions, one even containing a black hole, when the lapse is determined by a fifth ellipti...

We prove L 1 {contraction and thus uniqueness for solutions of the equation modelling un-stationary saturated{unsaturated water ow in porous media. Due to three diierent types of boundary conditions, describing contact with an impervious layer, with water reservoirs of variable level and with the atmosphere, this elliptic{parabolic equation comes as a varia-tional inequality with time{dependent...

We succeed in writing 2-dimensional conformally invariant nonlinear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the...

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equa...

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

where A is elliptic and B is monotone (not strictly). It is the case, for example, in porous medium, semiconductor equations, and so forth. In our application, we considered the injection moulding of a thermoplastic with a mold of small thickness with respect to its other dimensions. By averaging Navier-Stokes equations across the thickness of the mold and under an assumption (of Hele-Shaw) sta...

In this paper, we apply the Miura transformation to construct the connection between a variablecoefficient KdV (vcKdV) equation and a variable-coefficient modified KdV (vcmKdV) equation under certain constraint. Solving the vcmKdV equation by use of the auxiliary equation method and using the Miura transformation, we find a rich variety of new exact solutions for the vcKdV equation, which inclu...

We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ellitic form of the sixth Painlevé equation directly.

We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy speci c qualitative properties such as: (i) the norm of the gradient of one solution is bounded from below by a positive constant in the vicinity of a nite number of prescribed points; and (ii) the determinant of gradients of n solutions is bounded from...