Detecting non-uniqueness of solutions to biharmonic integral|equations through SVD

We consider the singular values of an integral operator and of a corresponding square matrix derived from the integral operator by means of a quadrature formula and a collocation. The integral operator and also the matrix depend on a real parameter, which may also enter the singular values of the operator and the matrix.

When a singular value drops to zero for a certain critical value of the parameter, the corresponding homogeneous integral equation or matrix equation has a nontrivial solution. Based on several examples with biharmonic integral operators we conjecture that the order of approximation of the critical value for the matrix is at least equal to the order of the quadrature formula used.

It is therefore possible – with a reasonable accuracy – to detect such critical values for the integral operator simply through a singular-value decomposition of the matrix derived by a quadrature and collocation.