Mixing cubic quasi-interpolation spline collocation method and optimal control techniques to solve polyharmonic (p=2 and p=3) problem

Abstract

This paper solves the polyharmonic equation for the cases $\mathrm{p\; =\; 2}$ and $\mathrm{p\; =\; 3}$, using an optimal control approach combined with the cubic quasi-interpolation spline collocation method. Specifically, the biharmonic and triharmonic problems are addressed by decomposing the high-order equation into a system of Poisson equations, which are then transformed into a minimization problem, following the principles of optimal control theory. The objective functional is constructed based on Neumann boundary conditions, while the constraints correspond to the Poisson equations resulting from the decomposition of the original problem. As the biharmonic case has been previously studied in Boudjaj et al. (2019), the main novelty of this work lies in the theoretical and numerical treatment of the triharmonic case. This case is reformulated as an optimal control problem, for which we prove the existence and uniqueness of the solution. Numerical experiments are carried out using the cubic quasi-interpolation spline collocation method. The results are compared with those obtained using the Localized Radial Basis Function (LRBFs) collocation method, highlighting the accuracy and efficiency of the proposed approach.