Abstract
A method for solving a general inverse coefficient problem for a Sturm–Liouville equation in impedance form \(\left(r^{2}(x)u^{\prime}(x)\right)^{\prime}+\lambda r^{2}(x)u(x)=0\) , \(x\in\left(0,L\right)\) , is proposed. It is based on Neumann series of Bessel functions (NSBF) representations for solutions of a related Schrödinger equation. The whole procedure reduces to a solution of a couple of systems of linear algebraic equations for the NSBF coefficients. Solving the first one, we recover a pair of characteristic functions of two Sturm–Liouville problems, while the solution of the second one leads to the knowledge of the NSBF coefficients on \(\left[0,L\right]\) , and in particular of the first NSBF coefficient \(g_{0}(x)\) . The unknown coefficient \(r(x)\) in the Sturm–Liouville equation is related with \(g_{0}(x)\) as \(r(x)=r(0)\left(g_{0}(x)+1\right)\) , that means that \(r(x)\) is recovered directly from the first component of the solution vector of the second system of linear algebraic equations. The approach leads to a simple and accurate numerical algorithm. Numerical efficiency is illustrated by test examples.
