The term fractional differentiation has recently been merged with the term fractal differentiation to create a new fractional differentiation operator. Several kernels were used to explore these new operators, including the power-law, exponential decay, and Mittag-Leffler functions. In this study, we analyze three forms of interpersonal relationships model. The numerical solution of the fractal–fractional interpersonal relationships model based on different kernels has been investigated. The new operators contain two parameters: one is for fractional order α, and the other is for fractal dimension γ. We use Lagrangian polynomial interpolation along with numerical method and the concept of fractional theory to solve these three forms of the titled model. All three forms of the numerical computation are compared with the solutions of the other existing method when α = γ = 1 that leads to a good agreement. The existence and uniqueness of the models have been studied using the Picard–Lindelöf theorem. To understand how the effects of fractal dimension and fractional order influence the model, we have illustrated various plots taking different values of α and γ.

中文翻译：

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具有幂、指数和米塔格-莱弗勒定律的人际关系的非线性分形-分数动态模型的数值解

最近，术语分数微分已与术语分形微分合并，以创建一个新的分数微分算子。使用了几个内核来探索这些新的算子，包括幂律、指数衰减和 Mittag-Leffler 函数。在本研究中，我们分析了三种形式的人际关系模型。研究了基于不同核的分形-分数人际关系模型的数值解。新的运算符包含两个参数：一个是分数阶α，另一个是分形维数γ. 我们使用拉格朗日多项式插值以及数值方法和分数理论的概念来解决这三种形式的标题模型。所有三种形式的数值计算都与其他现有方法的解决方案进行比较：α = γ = 1这导致了良好的协议。使用 Picard-Lindelöf 定理研究了模型的存在性和唯一性。为了了解分形维数和分数阶的影响如何影响模型，我们已经说明了采用不同值的各种图α和γ.