The delay PDEs are called partial functional differential equations as their unknown solutions are used in these equations as functional arguments. On the other hand, a neutral delay PDE is an especial case when the equation depends on the derivative(s) of the solution at some past stage(s). The current paper concerns to find an accurate and robust numerical solution for solving neutral delay time-space distributed-order fractional damped diffusion-wave equation based on the Galerkin meshless method. The test and trial functions for the used Galerkin method are constructed from the shape functions of reproducing kernel particle method (RKPM). For this aim, time derivative is approximated by a finite difference formula with convergence order $\mathcal{O}({\tau }^{3-\alpha })$ where $1<\alpha <2$. The stability and convergence of the time-discrete formulation are studied. For the next stage and to derive a fully-discrete scheme, we employ the Galerkin RKPM method. Moreover, the error estimate of the full-discrete scheme of new technique is discussed. Finally, an example is examined to check the proposed algorithm.

延迟偏微分方程称为偏函数微分方程，因为它们的未知解在这些方程中用作函数参数。另一方面，中性延迟 PDE 是一种特殊情况，当方程取决于某个过去阶段的解的导数时。当前论文关注的是找到一种基于伽辽金无网格方法的中性延迟时空分布阶分数阻尼扩散波方程的准确和稳健的数值解。所使用的伽辽金方法的测试和试验函数是由再生核粒子法（RKPM）的形状函数构建的。为此，时间导数近似为具有收敛阶数的有限差分公式$\mathcal{哦}({\tau }^{3-\alpha })$ 在哪里 $1<\alpha <2$. 研究了时间离散公式的稳定性和收敛性。对于下一阶段并导出完全离散的方案，我们采用 Galerkin RKPM 方法。此外，还讨论了新技术全离散方案的误差估计。最后，通过一个例子来检查所提出的算法。