Particular solutions of the multi-Helmholtz-type equation

This paper is devoted to finding analytic particular solutions to a class of fourth-order partial differential equations (PDEs). This is done by using polyharmonic spline approximations to the inhomogeneity. Both the 2-D and the 3-D cases are considered, the 3-D case being simpler than the 2-D one. The solutions to the 3-D case are obtained by using the Neumann expansion of the inverse of the homogeneous operator. For the 2-D case, in addition to the finding of the solutions to the inhomogeneous equation, it is necessary to find an appropriate basis for the radial form of the homogeneous equation. These solutions may have independent interest in obtaining t-Trefftz solutions to the homogeneous multi-Helmholtz-type equation.