This paper extends the recently proposed variant of meshless local Petrov Galerkin (MLPG) method, i.e., MLPG7, for solving time dependent PDEs. As test function, the method uses a novel modification of fundamental solution of Laplace operator that not only the test function itself but also its derivative vanish on boundary of local subdomains. Therefore, more stable local integral equations are obtained by replacing a boundary integral of derivatives of field variables with a domain integral of field variables itself. MLPG7 formulation converts the nonlinear governing equation into a system of first order ordinary differential equations (ODEs). For this system of ODEs, a theoretical stability analysis is carried out for the case of regularly spaced nodal points. Stability and convergence conditions for fully discretized equations by Euler (forward and backward) and Crank–Nicolson methods are investigated. The nonlinear term is treated iteratively. A numerical study investigates the convergence and stability of the method. A comparison is also done with some well-known methods and the results reveal the excellence of MLPG7. The convergence of the iterative approach is numerically verified by another illustrative test example. As nonlinear test problems, Allen–Cahn equations are studied.

中文翻译：

---

具有非线性反应项的扩散方程MLPG7的分析与应用

本文扩展了最近提出的无网格局部 Petrov Galerkin (MLPG) 方法的变体，即 MLPG7，用于求解时间相关的偏微分方程。作为测试函数，该方法使用拉普拉斯算子基本解的新颖修改，不仅测试函数本身而且其导数在局部子域的边界上消失。因此，用场变量本身的域积分代替场变量导数的边界积分，可以获得更稳定的局部积分方程。 MLPG7 公式将非线性控制方程转换为一阶常微分方程 (ODE) 系统。对于该常微分方程组，在节点间隔规则的情况下进行了理论稳定性分析。研究了欧拉（前向和后向）和 Crank-Nicolson 方法完全离散方程的稳定性和收敛条件。非线性项被迭代处理。数值研究研究了该方法的收敛性和稳定性。还与一些知名方法进行了比较，结果揭示了 MLPG7 的优越性。迭代方法的收敛性通过另一个说明性测试示例进行了数值验证。作为非线性测试问题，我们研究了 Allen–Cahn 方程。