Exponentially convergent symbolic algorithm of the functional-discrete method for the fourth order Sturm–Liouville problems with polynomial coefficients

Abstract

A new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the solution of fourth order Sturm--Liouville problem on a finite interval in the Hilbert space. The eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated. The sufficient conditions of an exponential convergence rate of the proposed approach are received. The obtained estimates of the absolute errors of FD-method significantly improve the accuracy of the estimates obtained earlier by I.P~Gavrilyuk, V.L.~Makarov and A.M.~Popov in 2010. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential coefficients and the correction number. Our method uses only the algebraic operations and basic operations on $(2\times 1)$ column vectors and $(2\times 2)$ matrices. The proposed approach does not require solving any boundary value problems and computations of any integrals, unlike the previous variants of FD-method by I.P.~Gavrilyuk, V.L.~Makarov, A.M.~Popov and N.M.~Romaniuk in 2010 and 2017. The corrections to eigenpairs are computed exactly as analytical expressions, and there are no rounding errors. The numerical examples illustrate the theoretical results. The numerical results obtained with the FD-method are compared with the numerical test results obtained with other existing numerical techniques.