In this work, we introduce a complete orthonormal (CON) set of functions as the eigenfanctions of a fourth-order boundary problem with radial symmetry. We derive the relation for the spectrum of the problem and solve it numerically. For larger indices n of the eigenvalues we derive accurate asymptotic representations valid within o{n^^). Two model fourth order problems with radial symmetry which admit exact analytic solutions are featured: a simple problem involving only the fourth-order radial operator and a constant and the other also involving the second-order radial operator. We show that for both cases, the rate of convergence is 0{N^^) which is compatible with theoretical predictions. The spectral and analytic solutions are found to be in excellent agreement. With 20 terms the absolute pointwise difference of the spectral and analytical solutions is of order 10^^ which means that the fifth order algebraic rate of convergence is fidly adequate.