Solving Sturm–Liouville inverse problems by an orthogonalized enhanced boundary function method and a product formula for symmetric potential

Abstract

We develop a fast convergence iterative method after transforming the Sturm–Liouville problem into the weak-form integral equations, and using sinusoidal functions as the testers as well as the bases of eigenfunction. Owing to the orthogonal property of these bases, we can obtain the expansion coefficients in closed-form. A new idea of orthogonalized and enhanced boundary function (OEBF) is introduced to compute eigenvalues. In the Sobolev space, we prove the closeness of the OEBF to the eigenfunction with their distance being reduced when the eigenvalue increases. Moreover, upon expressing the Rayleigh quotient in terms of OEBF, the unknown functions in the Sturm–Liouville operator can be recovered quickly, merely two boundary data of unknown functions and the first eigenvalue are sufficing. The OEBF is a promising method with a few iterations to obtain quite accurate estimations of the unknown potential and weight functions. Thanks to the symmetry property a symmetric matrix eigenvalue problem is derived to recover a symmetric potential. The characteristic equation endowing with a special coefficient matrix is decomposed into a product of two characteristic equations. The Newton iterative method is thus developed by relying on the product formula to reconstruct the unknown symmetric potential function with the aid of a few lower orders eigenvalues.