A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆2 ± λ2 (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, we applied the polyharmonic splines as the basis functionsto derive particular solutions for the differential operator ∆2 ± λ2. Similar to thederivation of fundamental solutions, it is non-trivial to derive particular solutionsfor higher order differential operators. In this paper, we provide a simple algebraicfactorization approach to derive particular solutions for these types of differentialoperators in 2D and 3D. The main focus of this paper is its simplicity in the sensethat minimal mathematical background is required for numerically solving higherorder partial differential equations such as thin plate vibration. Three numericalexamples in both 2D and 3D are given to validate particular solutions we derived.