The least square method for systems of linear ordinary differential equations (Q2337147)

The authors consider a boundary value problem for an overdetermined system of linear ordinary differential equations of first order. The given boundary conditions may also be redundant. Thus, the problem may have no solution. A variational approach is used to find generalized solutions which minimize some well-defined functionals. Two variants of the least square method are proposed. In the first one, the order of the equations is increased. Therefore, additional boundary conditions will be set. In the second variant, it is not necessarily assumed that the redundant boundary conditions are linear independent. In this case, the first technique cannot be applied. The case of coupled boundary conditions and implementation of the least square method by difference formulas are also investigated. Various (rather elementary) examples are given for the illustration and the comparison of the two variants of the least square methods. The techniques that are proposed in this paper are particularly useful in the situation when the original data is perturbed so that the boundary value problem may have no solution.