In this article, we investigate the generalized fractional operator Caputo type (ABC) with kernels of Mittag–Lefller in three parameters and its fractional integrals with arbitrary order for solving the time fractional parabolic nonlinear equation. The generalized definition generates infinitely many problems for a fixed fractional derivative α. We utilize this operator with homotopy analysis method for constructing the new scheme for generating successive approximations. This procedure is used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the ℏ-curves as well as by calculating the residual error and the results were accurate. Based on the experiment, we verify the existence of the solution for the new parameters. Depending on these results, this treatment can be used to find approximate solutions to many fractional differential equations.

中文翻译：

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具有广义 Mittag-Leffler 核的分数抛物型方程的数值解

在本文中，我们研究了具有三参数 Mittag-Lefller 核的广义分数算子 Caputo 型（ABC）及其任意阶分数积分，用于求解时间分数抛物线非线性方程。对于固定的分数阶导数α ，广义定义会产生无限多个问题。我们利用该算子和同伦分析方法来构建生成逐次逼近的新方案。此过程已成功用于两个示例以找到解决方案。通过明确ℏ曲线的收敛区域以及计算残差来验证该方法的有效性和准确性，结果是准确的。基于实验，我们验证了新参数解的存在性。根据这些结果，这种处理可用于找到许多分数阶微分方程的近似解。