This paper introduces general methodologies for constructing closed-form solutions to several important partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. The covered equations include the isotropic and anisotropic Poisson, Helmholtz, Stokes, and elastostatic equations, as well as the time-harmonic linear elastodynamic and Maxwell equations. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and have potential applications in certain kinds of Trefftz finite element methods. Our approach to all of these PDEs relates the particular solution to polynomial solutions of the Poisson and Helmholtz polynomial particular solutions, solutions that can in turn be obtained, respectively, from expansions using homogeneous polynomials and the Neumann series expansion of the operator $(k^2+\Delta)^{-1}$. No matrix inversion is required to compute the solution. The method naturally incorporates divergence constraints on the solution, such as in the case of Maxwell and Stokes flow equations. This work is accompanied by a freely available Julia library, \texttt{PolynomialSolutions.jl}, which implements the proposed methodology in a non-symbolic format and efficiently constructs and provides access to rapid evaluation of the desired solution.

翻译：本文提出了在二维和三维空间中构造几类重要偏微分方程（PDEs）闭式解的一般方法，这些方程包含多项式右端项。涵盖的方程包括各向同性和各向异性的泊松方程、亥姆霍兹方程、斯托克斯方程、弹性静力学方程，以及时谐线性弹性动力学和麦克斯韦方程组。多项式解近年来在用于评估体积积分算子的数值技术发展中重新获得重要地位，并在某些特雷夫茨有限元方法中具有潜在应用。针对所有这些偏微分方程，我们的方法将特解与泊松方程和亥姆霍兹方程的多项式特解相关联，而这些多项式特解可分别通过齐次多项式的展开和算子 $(k^2+\Delta)^{-1}$ 的诺伊曼级数展开求得。求解过程无需矩阵求逆。该方法自然地纳入了对解的散度约束，例如在麦克斯韦和斯托克斯流方程中。本工作附带一个免费提供的 Julia 库 \texttt{PolynomialSolutions.jl}，该库以非符号形式实现了所提出的方法，并能高效地构建和快速评估所需解。