LAUSR

Abstract For a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations. Research Highlights – Two novel algorithms are developed for a second-order nonlinear singularly perturbed boundary value problem which automatically satisfies the Robin boundary conditions. – The new idea of boundary shape function is first introduced, whose existence is proven and it can automatically satisfy the Robin boundary conditions. – In the first type algorithm the free functions are the exponential type bases endowed with fractional powers, which satisfy the Robin boundary conditions automatically, and can capture the singular behavior. – In the second one we let the free function be another variable, such that the boundary value problem can be transformed to an initial value problem for the new variable, which can quickly find accurate solution for the nonlinear problem through a few iterations.