On Solutions for a Generalized Differential Equation Arising in Boundary Layer Problem

We treat the existence and uniqueness of a solution for the generalized Blasius problem which arises in boundary layer theory. The shooting method is used in the proof of our main result. An example is included to illustrate the results. 1. Introduction The steady motion in the boundary layer along a thin flat plate which is immersed at zero incidence in a uniform stream with constant velocity can be described [1] in terms of the solution of the differential equation: which satisfies the boundary conditions This problem was first solved numerically by Blasius [2] and is the subject of a vast literature. Some generalizations of the Blasius equation can be found in [3–5]. In [3], the authors investigate the model , , arising in the study of a laminar boundary layer for a class of non-Newtonian fluids. In [4], the author considers the equation , which describes boundary layer flows with temperature dependent viscosity. It is our goal to study the existence of solutions to the generalized boundary value problem consisting of the nonlinear third order differential equation subject to the boundary conditions (2). We assume that the functions , , and are continuous. The additional conditions imposed on and in (3) are the following ones:(H1) for some and all ;(H2) ;(H3) for some and all . An example of (3), which satisfies the conditions (H1), (H2), and (H3), is . If and , then (3) coincides with the Blasius equation (1). For the related Falkner-Skan equation [6] the similar generalization was given in [7]. Falkner-Skan equation describes the steady two-dimensional flow of a slightly viscous incompressible fluid past a wedge of angle ( ). The shooting method [8] is used for treating the existence and the number of solutions to boundary value problem. The shooting method reduces solving a boundary value problem to solving of an initial value problem. So we consider solution of the auxiliary initial value problem for (3) with initial data and we are looking for and such that and . Applying the intermediate value theorem, continuity of with respect to leads to the existence of at least one such that . The paper is organized as follows. Section 2 contains some auxiliary results. Section 3 is devoted to the properties of solutions of initial value problem (3), (4). In Section 4 we consider dependence of solutions on initial data. In Section 5 we deal with solutions to boundary value problem (3), (2). Also one example is given to illustrate the results. The ideas of the proofs of some results are taken from [6]. 2. Preliminary Results Proposition 1. Suppose that a