This work presents an iterative scheme for the numerical solution of the space‐time fractional two‐dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional‐order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two‐dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two‐dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two‐dimensional nonlinear mathematical models for various situations.

中文翻译：

---

二维时空分数扩散方程的近似解析解

这项工作为使用Caputo分数阶导数的Laplace变换应用同伦扰动的时空分数维对流-反应-扩散方程的数值解提供了一个迭代方案。所获得的解对于分析超扩散系统和亚扩散系统的建模，异常扩散，在多孔介质中的传输过程是有益和有意义的。该迭代技术提出了同伦摄动技术以及Heplace多项式的Laplace变换的组合，可以将其进一步应用到许多线性/非线性二维分数模型中，以计算近似解析解。在本方法中，非线性可以通过He多项式解决。