Inverse problems for discrete Hermite nabla difference equation

Inverse problems are studied for discrete Hermite equations with nabla difference including initial value, terminal value and Sturm–Liouville problems. A quantitative study is conducted to obtain the solution. The problem with initial value and terminal value conditions is transformed into a simple recursive formula while the Sturm–Liouville problem is transformed into a generalized eigenvalue problem. Some analyses related to the corresponding matrix are done. The eigenvalues are computed using the determinant of the corresponding pencil matrix. Furthermore, a simple and fast recursive algorithm is proposed for computing corresponding eigenvectors. Several examples are provided to explain the method simply and to demonstrate the numerical results.