我們提出了一種線性分數階微分方程的新表示法。此類分數階微分方程常用以描述在無限制承載量下的族群動態行為，其中分數階數為一參數，用以刻劃由於環境異質性或結構性限制所導致的物種遷徙或繁殖速率下降的情形。

本研究採用的方法為首次提出，其靈感來自於解常微分方程時所使用的技巧。我們透過一種迭代機制來構造這個分數階微分方程的解。在此機制下，我們發現由迭代過程所產生的一系列函數因具備週期性特徵而為有限個。此外，我們亦證明每一個迭代所得的函數皆為初始函數的分數階微分，據此可將分數階微分方程的解表示為這些函數的線性組合。

事實上，我們所得的解可視為米塔-列夫勒函數的重新排列形式，而該函數為本方程問題中的唯一解。然而，透過將解表達為有限項解析函數的總和，我們得以更有效地分析其漸近行為。

We find a new representation of the linear fractional differential equation (FDE). This FDE is often used to describe the population dynamic with unlimited carrying capacity, and the fractional order is the parameter that describes species’ migration or reproduction slowed down by environmental heterogeneity or structural limitations.

Our approach is proposed for the first time, inspired by techniques used in solving ordinary differential equations (ODE). We use an iterative scheme to construct the solution of the FDE. Under this scheme we discover that the series of functions generated by iteration process is finite due to the periodic property. Moreover, we also prove that each iterated functions is the fractional derivative of the first function, so that we can write the solution as a linear combination of these functions.

Our solution is in fact a rearrangement of the Mittag-Leffler function, which is known as the unique solution of the FDE in our problem. However, by writing the solution into a finite-term sum of analytic functions, we can analyze the asymptotic behavior more efficiently.

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