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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. [ABSTRACT FROM AUTHOR]

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods. [ABSTRACT FROM AUTHOR]