Fractional Differential Equations: Numerical Methods for Applications

This book provides a comprehensive set of practical tools for exploring and discovering the world of fractional calculus and its applications, and thereby a means of bridging the theory of fractional differential equations (FDE) with real-world facts. These tools seamlessly blend centuries old numerical methods such as Gaussian quadrature that have stood the test of time with pioneering concepts such as hypermatrix equations to harness the emerging capabilities of modern scientific computing environments. This unique fusion of old and new leads to a unified approach that intuitively parallels the classic theory of differential equations, and results in methods that are unprecedented in computational speed and numerical accuracy.

The opening chapter is an introduction to fractional calculus that is geared towards scientists and engineers. The following chapter introduces the reader to the key concepts of approximation theory with an emphasis on the tools of numerical linear algebra. The third chapter provides the keystone for the remainder of the book with a comprehensive set of methods for the approximation of fractional order integrals and derivatives. The fourth chapter describes the numerical solution of initial and boundary value problems for FDE of a single variable, both linear and nonlinear. Moving to two, three, and four dimensions, the ensuing chapter is devoted to a novel approach to the numerical solution of partial FDE that leverages the little-known one-to-one relation between partial differential equations and matrix and hypermatrix equations. The emphasis on applications culminates in the final chapter by addressing inverse problems for ordinary and partial FDE, such as smoothing for data analytics, and the all-important system identification problem.

Over a century ago, scientists such as Ludwig Boltzmann and Vito Volterra formulated mathematical models of real materials that -- based on physical evidence -- integrated the history of the system. The present book will be invaluable to students and researchers in fields where analogous phenomena arise, such as viscoelasticity, rheology, polymer dynamics, non-Newtonian fluids, bioengineering, electrochemistry, non-conservative mechanics, groundwater hydrology, NMR and computed tomography, mathematical economics, thermomechanics, anomalous diffusion and transport, control theory, supercapacitors, and genetic algorithms, to name but a few. These investigators will be well-equipped with reproducible numerical methods to explore and discover their particular field of application of FDE.

本書提供了一套全面的實用工具，用於探索和發現分數微積分及其應用的世界，從而將分數微分方程（FDE）的理論與現實世界的事實相結合。這些工具無縫地結合了經過時間考驗的數百年的數值方法（如高斯積分）和開創性的概念（如超矩陣方程），以利用現代科學計算環境的新興能力。這種新舊融合的獨特方法直觀地類比了經典微分方程理論，並產生了在計算速度和數值精度方面前所未有的方法。

第一章是針對科學家和工程師的分數微積分介紹。接下來的一章介紹了近似理論的關鍵概念，重點介紹了數值線性代數的工具。第三章提供了一套全面的方法，用於近似分數階積分和導數。第四章描述了單變量FDE的初始值和邊界值問題的數值解，包括線性和非線性問題。接下來的章節涉及到二、三和四維度，專注於利用部分微分方程與矩陣和超矩陣方程之間鮮為人知的一對一關係的新方法來數值解決部分FDE。最後一章強調應用，解決了普通和部分FDE的反問題，例如數據分析的平滑和重要的系統識別問題。

一個多世紀以前，像路德維希·玻爾茨曼和維托·沃爾特拉這樣的科學家制定了基於物理證據的真實材料的數學模型，這些模型整合了系統的歷史。本書對於在類似現象出現的領域中的學生和研究人員將是無價之寶，例如粘彈性、流變學、聚合物動力學、非牛頓流體、生物工程、電化學、非保守力學、地下水水文學、核磁共振和計算機斷層掃描、數學經濟學、熱力學力學、異常擴散和傳輸、控制理論、超級電容器和遺傳算法等等。這些研究人員將具備可重現的數值方法，以探索和發現分數微分方程在他們特定的應用領域中的潛力。