The Method of Polynomial Particular Solutions for Solving Nonlinear Poisson-Type Equations

In this paper, the method of polynomial particular solutions is used to solve nonlinear Poisson-type partial differential equations in one, two, and three dimensions. The condition number of the coefficient matrix is reduced through the implementation of multiple scale technique, ultimately yielding a stable numerical solution. The methodological process can be divided into two main parts: first, identifying the corresponding polynomial particular solutions for the linear differential operator terms in the governing equations, and second, employing these polynomial particular solutions as basis function to iteratively solve the remaining nonlinear terms within the governing equations. Additionally, we investigate the potential improvement in numerical accuracy for equations with singularities in the analytical solution by shifting the computational domain a certain distance. Numerical experiments are conducted to assess both the accuracy and stability of the proposed method. A comparison of the obtained results with those produced by other numerical methods demonstrates the accuracy, stability, and efficiency of the proposed method in handling nonlinear Poisson-type partial differential equations.