Solution of a Two-Point Boundary Value Problem with Jacobian Matrix Characterized by Extremely Large Eigenvalues,
Abstract
The paper treats the nonlinear, two-point boundary-value problem d squared x/dt squared - k sinh(kx) = 0, x(0) = 0, x(1) = 1 for relatively large values of k, namely, k = 5, k = 6, and k = 10. Computationally speaking, this is an extremely difficult problem, owing to the fact that the Jacobian matrix is characterized by an extremely large positive eigenvalue: for k = 10, the order of magnitude of this positive eigenvalue near the final point is 1,000. The resulting numerical difficulties are reduced by treating the two-point boundary-value problem as a multipoint boundary-value problem. The modified quasilinearization algorithm is employed. This is a totally finite- difference approach, which bypasses the integration of the nonlinear equations, which characterizes shooting methods. Solutions for x(t) precise to six significant figures are obtained. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1972
- Accession Number
- AD0761788
Entities
People
- A. K. Aggarwal
- Angelo Miele
- J. L. Tietze
Organizations
- Rice University