In this article, we suggest a numerical approach based on q‐homotopy analysis Elzaki transform method (q‐HAETM) to solve fractional multidimensional diffusion equations which represents density dynamics in a material undergoing diffusion. We take the noninteger derivative in the Caputo–Fabrizio kind. The proposed method, q‐HAETM is an advanced adaptation in q‐HAM and Elzaki transform method which makes mathematical calculation very effective additionally more accurate. Since, in classical perturbation scheme, the scheme restricted to the small parameter whereas the q‐HAETM is not restricted to the small parameter. By theoretical and numerical evaluation it is observed that q‐HAETM yields an analytical solution in the form of a convergent series. By taking three examples and applying q‐HAETM, the numerical results reveal that the suggested method is straightforward to apply and computationally very effective.

中文翻译：

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带指数记忆的分数维扩散方程的高效数值方法

在本文中，我们建议基于q同伦分析Elzaki变换方法（q‐ HAETM）的数值方法来解决分数多维扩散方程，该方程表示正在扩散的材料中的密度动力学。我们采用Caputo–Fabrizio类型的非整数导数。所提出的方法q- HAETM是q- HAM和Elzaki变换方法的高级改进，这使得数学计算非常有效，而且更加准确。因为在经典摄动方案中，该方案仅限于小参数，而q- HAETM则不受限于小参数。通过理论和数值评价，观察到Q-HAETM提供收敛系列的分析解决方案。通过三个示例并应用q- HAETM，数值结果表明，所建议的方法易于直接应用，并且在计算上非常有效。