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Method of particular solutions for nonlinear Poisson-type equations in irregular domains

机译：不规则域中非线性泊松型方程的特解方法

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摘要

The paper presents a new meshless numerical technique for solving nonlinear Poisson-type equations V~2u = F(u,u_(x_1), ,u_(x_2),x)+f(x) in irregular domains. Using the dual reciprocity method (DRM) approach, the nonlinear term is represented as a linear combination of basis functions F ≌ ∑~M_m q_mφ_m.We use RBFs as the basis functions φ_m. According to the DRM this admits of writing the approximate solution in the form of the linear combination of particular solutions corresponding to these basis functions u_M = u_f+ ∑~M_mq_mφ_m We use tne particular solution as a sum of two parts Φ = Φ~0 + Φ~1, where Φ~0 is the well known analytical solution of the Poissone equation with a RBF on the right-hand side. The second term Φ~1 provides that Φ satisfies the homogeneous conditions on the boundary of the domain. As a result, it is possible to isolate the nonlinear part of the equation and the problem is reduced to a system of nonlinear equations F(u_m,u_(m,x_1),u_(m,x_2),x) = ∑~M_mq_mφ_m for tne unknown coefficients q_m. Then the nonlinear system is solved numerically. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions or other numerical methods indicates that the proposed method is accurate and efficient in dealing with complicated geometry and strong nonlinearity.
机译：本文提出了一种新的无网格数值技术，用于求解不规则域中的非线性泊松型方程V〜2u = F（u，u_（x_1），，u_（x_2），x）+ f（x）。使用对等互易（DRM）方法，将非线性项表示为基函数F≌∑〜M_mq_mφ_m的线性组合。我们使用RBF作为基函数φ_m。根据DRM，这允许以对应于这些基本函数的特定解的线性组合的形式写近似解u_M = u_f + ∑〜M_mq_mφ_m我们使用特定解作为两个部分的总和Φ=Φ〜0 +Φ 〜1，其中Φ〜0是右边带RBF的Poissone方程的众所周知的解析解。第二项Φ〜1规定Φ满足域边界上的齐次条件。结果，可以隔离方程的非线性部分，并将问题简化为非线性方程组F（u_m，u_（m，x_1），u_（m，x_2），x）= ∑〜M_mq_mφ_m对于未知系数q_m。然后对非线性系统进行数值求解。进行数值实验以进行准确性和收敛性研究。将本文获得的数值结果与精确解或其他数值方法进行比较表明，该方法在处理复杂的几何形状和强非线性方面是准确而有效的。

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