In this paper, we proposed an asymptotic approach with respect to a small parameter for fractional differential equations, with the small parameter linked to the fractional order derivative. This approach allows the splitting of the field variable and consequently the Riemann–Liouville Integral as the sum of two contributions. One describes the unperturbed state and the other describes the behavior of the model in the perturbed state due to the presence of the fractional derivative. This approach allows mapping time-fractional differential equations into a system of two coupled partial differential equations with integer order. By solving these partial differential equations we are able to obtain solutions of the assigned fractional differential equations. To obtain solutions we determine the approximate Lie symmetries through which the system of two coupled partial differential equations is reduced to a system of two coupled ordinary differential equations. As an application, we consider the time fractional diffusion–reaction equation with a linear source term; we test the proposed procedure by comparing the profiles of the obtained approximate solutions with solutions found in the fractional context for values of the fractional parameter tending to the integer value.

中文翻译：

---

黎曼-刘维尔导数分数阶微分方程小参数渐近展开法


在本文中，我们提出了一种关于分数阶微分方程小参数的渐近方法，其中小参数与分数阶导数相关。这种方法允许将场变量分裂，从而将黎曼-刘维尔积分分裂为两个贡献的总和。一种描述未扰动状态，另一种描述由于存在分数导数而导致模型在扰动状态下的行为。这种方法允许将时间分数阶微分方程映射到两个整数阶耦合偏微分方程组中。通过求解这些偏微分方程，我们可以获得指定的分数阶微分方程的解。为了获得解，我们确定近似李对称性，通过该近似李对称性，两个耦合偏微分方程组被简化为两个耦合常微分方程组。作为一个应用，我们考虑具有线性源项的时间分数扩散反应方程；我们通过将获得的近似解的轮廓与在分数上下文中找到的分数参数值趋于整数值的解进行比较来测试所提出的过程。