A revisit on the derivation of the particular solution for the differential operator $\Delta^2 \pm \lambda^2$

Abstract

In this paper, we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator Δ2 ± λ2. Similar to the derivation of fundamental solutions, it is non-trivial to derive particular solutions for higher order differential operators. In this paper, we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D. The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration. Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.