The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

帕斯捷尔纳克地基上的简单支撑薄板弯曲问题的控制方程通过中间变量被分解为两个耦合的低阶微分方程，分别是亥姆霍兹方程和拉普拉斯方程。提出了一个二维Helmholtz算子的新解，如附录1所示。使用二维Helmholtz算子和Laplace算子的R函数和基本解来构造相应的准Green函数。拟格林函数满足问题的齐次边界条件。应用相应的格林公式，算子的基本解和边界条件，将亥姆霍兹方程和拉普拉斯方程转换为积分方程。构造新的边界归一化方程以确保积分核的连续性。积分方程被离散化为非齐次线性代数方程，以进行数值计算。数值算例验证了该方法在简单边界条件和多边形边界条件下计算问题的有效性。所需的结果是通过MATLAB编程获得的。讨论了该方法的收敛性。与解析解的比较表明吻合良好，证明了本文方法的可行性和有效性。数值算例验证了该方法在简单边界条件和多边形边界条件下计算问题的有效性。所需的结果是通过MATLAB编程获得的。讨论了该方法的收敛性。与解析解的比较表明吻合良好，证明了本文方法的可行性和有效性。数值算例验证了该方法在简单边界条件和多边形边界条件下计算问题的有效性。所需的结果是通过MATLAB编程获得的。讨论了该方法的收敛性。与解析解的比较表明吻合良好，证明了本文方法的可行性和有效性。