An invariant imbedding method for computation of eigenlengths of singular two-point boundary-value problems

In this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular TPBVP with the singularity at the origin. The invariant imbedding yields a first-order nonlinear equation called a Riccati equation and also gives the initial conditions at the origin for this equation. With the aid of Theorem 8 in Section 3 we numerically integrate the Riccati equation to “blowup” which gives our computed eigenlength.In closing, we would like to comment on the numerical merits of the integration-to-blowup technique. On the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results. The feature of a variable step size is essential as the value of z approaches the actual eigenlength. However, it is desirable to have a priori estimate or bounds of the eigenlength similar to those of Boland and Nelson 2] for the nonsingular case. The singular system, however, presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds. Hopefully an investigation of the matrix R(z) will yield these results.