The Compound Matrix Method for Multi-Point Boundary-Value Problems Depending on a Parameter

Abstract

Boundary-value problems for ordinary differential-equation (ODE) systems often depend on a parameter. It is shown how derivatives of the solution with respect to the parameter can be computed using compound matrices in the linear multi-point case. In the applications, such derivatives are useful e.g. in inversion theory when the parameter is to be estimated. When the dependence on the parameter is analytic, an integral of boundary-problem solutions with respect to the parameter can typically be expanded as a sum of residues. Such integrals and expansions have theoretical as well as practical interest, and an explicit formula is derived for the residue contributions. It is given in terms of eigensolutions to the original problem and an appropriately defined adjoint problem. It is shown how the quantities involved can be computed in a stable way using compound matrices. For the application case with wave-propagation in a range-independent multi-region fluid-solid medium, a solution to the adjoint problem is obtained directly from a solution to the original problem, and the well-known formula for modal excitation coefficients is extended to leaky modes. Modal depth functions can be computed reliably without experimentation with a cut-off depth for an artificial homogeneous half-space.