Solving a nonlinear inverse Sturm–Liouville problem with nonlinear convective term using a boundary functional method

Abstract

This article is concerned with solving nonlinear inverse problems to recover the second-order nonlinear Sturm–Liouville operators, probably including a nonlinear convective term, which refers to over-specified boundary data. Utilising a homogenization approach with the boundary data, a series of boundary functions are obtained and consist of a linear space combined with the zero element. The energetic functional both qualifying the homogeneous boundary conditions and maintaining the energy is presented in the linear space based on the energetic boundary functions. The multiplier can be decided by solving a nonlinear equation. The linear systems utilized in rebuilding the uncertain leading coefficient function as well as the potential function in the nonlinear Sturm–Liouville operator are improved, so that the iterative algorithms converge rapidly. Numerical examples provided indicate that the presented method is well capable of recovering the nonlinear Sturm–Liouville operators with a nonlinear convective term.