Many classes of singularly perturbed reaction-diiusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as no...

We show that the Painlevé equations PIII −PV I can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrödinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three different branches, each described by an infinite chain of equations. The Painlevé equations are obtained by closing the chain periodically at the lowe...

This paper deals with the eigenvalue singular problem of the spectral type differential equations of the fourth-order self-adjoint differential operators 1 ð1 xÞð1þ xÞ d dx ð1 xÞð1þ xÞ d y dx " # ly 1⁄4 0; yð 1Þ; yð1Þ finite where, p P 1; q P 1. It has been reported in the literature that spectral type differential equations above have general solutions in terms of hypergeometric functions. In ...

A new code is presented for solving linear eigenvalue problems from fluid models of the edge plasma of tokamaks. The 2DX code solves linearized fluid equations in a 2D cross-section of the plasma, with toroidal mode number resolving the third dimension. Geometry capabilities include both closed and open field lines, allowing solution of x-point problems as well as a variety of other toroidal an...

This study aimed at linear stability analysis of the stratified two-phase liquid-gas flow in a horizontal pipe. First, equations governing the linear stability of flow in each phase and boundary conditions were obtained. The governing equations were eigenvalue Orr Sommerfeld equations which are difficult and stiff problems to solve. After obtaining the velocity profiles of the gas and liquid ph...

Our aim is to provide the open domain MATLAB code bvpsuite for the efficient numerical solution of boundary value problems in ordinary differential equations. Motivated by applications, we are especially interested in designing a code whose scope is appropriately wide, including fully implicit problems of mixed orders, parameter dependent problems, problems with unknown parameters, problems pos...

We investigate a D dimensional generalization of the Schroedinger-Newton equations, which purport to describe quantum state reduction as resulting from gravitational effects. For a single particle, the system is a combination of the Schroedinger and Poisson equations modified so that the probability density of the wavefunction is the source of the potential in the Schroedinger equation. For sph...

Radial Basis Function (RBF) collocation methods for time-dependent PDEs, in particular hyperbolic PDEs, are known to be difficult to implement in a way so that they are stable for time integration. It has been hypothesized that the instability is due to the way that boundary conditions are applied and to the relatively large errors in boundary regions. We describe a preconditioning technique th...

We consider elliptic equations in bounded domains Ω ⊂ R with nonlinearities which have exponential growth at +∞ (subcritical and critical growth respectively) and linear growth λ at −∞, with λ > λ1, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical gro...

We analyze a preconditioner for the time-independent Wigner-Poisson equations for a resonant tunneling diode and present a numerical example of a continuation study which supports the theory. The application of the preconditioner transforms the equations into a compact fixed point problem for the Wigner distribution. After discretization, the Jacobians of the discrete problem have mesh-independ...