An approximate solution for homogeneous boundary-value problems with slowly-varying coefficient matrices (Q1336884)

An approximate closed-form solution is obtained for linear homogeneous boundary-value problems with slowly-varying coefficient matrices. The derivation of the approximation solution is based on the freezing technique, which once solves the problem with the matrices ``frozen'' at a certain point and then relaxes it so as to have the independent variable. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the problem. The proposed approximate solution is compared to the numerical solution in a two-point boundary-value problem. Good agreement is observed between two solutions when the rate of change of the matrix is small.