Time-Fractional Order Biological Systems with Uncertain Parameters

The subject of fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its validated applications in various fields of science and engineering. It is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The fractional derivative has been used in various physical problems, such as frequency-dependent damping behavior of structures, biological systems, motion of a plate in a Newtonian fluid, ����λ��μ controller for the control of dynamical systems, and so on. It is challenging to obtain the solution (both analytical and numerical) of related nonlinear partial differential equations of fractional order. So for the last few decades, a great deal of attention has been directed towards the solution for these kind of problems. Different methods have been developed by other researchers to analyze the above problems with respect to crisp (exact) parameters.

However, in real-life applications such as for biological problems, it is not always possible to get exact values of the associated parameters due to errors in measurements/experiments, observations, and many other errors. Therefore, the associated parameters and variables may be considered uncertain. Here, the uncertainties are considered interval/fuzzy. Therefore, the development of appropriate efficient methods and their use in solving the mentioned uncertain problems are the recent challenge.

In view of the above, this book is a new attempt to rigorously present a variety of fuzzy (and interval) time-fractional dynamical models with respect to different biological systems using computationally efficient method. The authors believe this book will be helpful to undergraduates, graduates, researchers, industry, faculties, and others throughout the globe.

分數微積分的主題在過去三十年間獲得了相當大的流行度和重要性，主要是由於它在科學和工程的各個領域中得到了驗證的應用。它是對任意（非整數）階數的普通微分和積分的一種概括。分數導數已被應用於各種物理問題，例如結構的頻率依賴阻尼行為、生物系統、牛頓流體中板的運動、用於動態系統控制的λμ控制器等。獲得分數階偏微分方程的解（包括解析和數值解）是具有挑戰性的。因此，在過去幾十年中，人們對解決這類問題給予了很大的關注。其他研究人員已經開發了不同的方法來分析上述問題，以考慮到精確（確定）參數。

然而，在生物問題等實際應用中，由於測量/實驗誤差、觀察誤差和其他錯誤，往往無法獲得相關參數的精確值。因此，相關參數和變量可能被視為不確定的。在這裡，不確定性被視為區間/模糊。因此，開發適當高效的方法並將其應用於解決上述不確定問題是當前的挑戰。

基於上述情況，本書是一次嚴謹地介紹使用計算高效方法對不同生物系統的模糊（和區間）時間分數動力學模型的新嘗試。作者相信本書將對全球的本科生、研究生、研究人員、工業界、教職員工和其他人有所幫助。