Here, we generalize the boundary layer functions method (or composite asymptotic expansion) for bisingular perturbed differential equations (BPDE that is perturbed differential equations with singular point). We will construct a uniform valid asymptotic solution of the singularly perturbed first-order equation with a turning point, for BPDE of the Airy type and for BPDE of the second-order with a regularly singular point, and for the boundary value problem of Cole equation with a weak singularity.A uniform valid expansion of solution of Lighthill model equation by the method of uniformization and the explicit solution-this one by the generalization method of the boundary layer function-is constructed. Furthermore, we construct a uniformly convergent solution of the Lagerstrom model equation by the method of fictitious parameter.

No abstract

No abstract

No abstract

where 0 < ~ << 1, k E N, and/3 k 0 is a numerical parameter. This problem was proposed by Lagerstrom as the model problem for the Navier-Stokes equation with small Reynolds numbers. The problem can be treated as a problem of the distribution of a stationary temperature v(r).The first two terms in (1) represent the (k + 1)-dimensional Laplacian, dependent only on the radius, and the other two terms reflect nonlinear heat losses.The asymptotics of a solution of the problem (1) was previously studied by the method of joining [1, 2] and by the method of two-sided approximations [3].It turns out that not only an asymptotic solution but also the convergent solution of Eq.(1) are constructed very simply by the fictitious-parameter method [4]. The principal idea of this method consists in the following. For the initial problem a fictitious parameter A E [0, 1] is introduced such that:(1) for A = 0 the solution of Eq.(1) satisfies all initial or boundary conditions; (2) for all A E [0, 1] the solution of problem (1) is expanded in a series in integer or fractional powers of A . It is convenient to make in Eq. (1) the substitution r = Cx, v = 1 -u. Then problem (1) can be rewritten as