Pseudospectral solutions to some singular nonlinear BVPs

Abstract

We solve by Laguerre collocation method two second order genuinely nonlinear singularly perturbed boundary value problems and a fourth order nonlinear one formulated on the half-line. First, we are concerned with solutions to a model equation for low Reynolds number flow, the so-called Cohen, Fokas, Lagerstrom model. Then we compute monotonously increasing solutions, the so-called “bubble type solutions”, to density profile equation for the description of the formation of microscopical bubbles in a nonhomogeneous fluid. We compare the results obtained in the latter case with those carried out by shooting or polynomial collocation both coupled with domain truncation. A fourth order nonlinear boundary value problem supplied with mixed boundary conditions in origin as well as at infinity is also solved. The solution and its first three derivatives are obtained and compared with their counterparts obtained by Keller’s box scheme. The collocation based on Laguerre functions avoids the domain truncation, exactly impose any kind of boundary conditions at infinity and resolve with reliable accuracy the sharp interior or exterior layer of the solutions. The method is fairly easy implementable and efficient in treating various types of nonlinearities. The accuracy of the method is also tested on the classical Blasius problem (a third order one). We solve with the same method the eigenvalue problem obtained by linearization around the constant solution, which corresponds to the case of a homogeneous fluid (without bubbles), and observe that this solution is stable.