In this article, we introduced for the first time the two-step Adomian decomposition method (TSADM) for solving the multi-dimensional Riesz space distributed-order advection–diffusion (RSDOAD) equation. The TSADM was successfully applied to obtain the analytical solution of the multi-dimensional (RSDOAD) equation. The analytical solution has been obtained without approximation/discretization of the Riesz fractional operator. Furthermore, new results for the existence are obtained with the help of some fixed point theorems, while the uniqueness of the solution was investigated employing the Banach contraction principle. Finally, we included a generalized example to demonstrate the validity and application of the proposed method. The obtained results conclude that the proposed method is powerful and efficient for the considered problem compared to the other existing methods.

中文翻译：

---

两步Adomian分解法求解多维Riesz空间分布阶次对流-扩散方程的存在唯一性结果及解析解

在本文中，我们首次介绍了求解多维 Riesz 空间分布式阶次对流扩散 (RSDOAD) 方程的两步 Adomian 分解方法 (TSADM)。TSADM 成功应用于获得多维 (RSDOAD) 方程的解析解。解析解是在没有对 Riesz 分数算子进行近似/离散化的情况下获得的。此外，在一些不动点定理的帮助下，获得了存在性的新结果，同时利用巴拿赫收缩原理研究了解的唯一性。最后，我们包含了一个通用示例来证明所提出方法的有效性和应用。