A novel method for a class of nonlinear singular perturbation problems
The problem under consideration is the singularly perturbed boundary value problem \[ \varepsilon y'' + f(x,y)y' + g(x,y) = 0, \quad 0 < x < 1, \quad y(0) = \alpha , \quad y(1) = \beta \] The solution of the problem is constructed as the sum \[ y(x) = U(x) + V(x/\varepsilon) + O(\varepsilon ). \] The regular part \(U\) is defined by the initial value problem \[ f( x,U )U' + g(x,U) = 0,\quad U(1) = \beta , \] and the boundary layer function \(V\) is defined by the correction problem \[ V'' + f(0,U(0) + V)V' = 0, \quad V(0) = \alpha - U(0), \quad \lim_{t \to \infty } V(t) = 0. \] To solve the last problem, the author suggests a shooting technique based on the series method and Padé approximation. Numerical experiments are presented.
scientific article; zbMATH DE number 2118013