商品描述
Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: •Vol 1: Introduction to Algorithms and Computer Coding in R •Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. In the second volume, the emphasis is on applications of SFPDEs developed mainly through the extension of classical integer PDEs to SFPDEs. The example applications are: •Fractional diffusion equation with Dirichlet, Neumann and Robin boundary conditions •Fisher-Kolmogorov SFPDE •Burgers SFPDE •Fokker-Planck SFPDE •Burgers-Huxley SFPDE •Fitzhugh-Nagumo SFPDE. These SFPDEs were selected because they are integer first order in time and integer second order in space. The variation in the spatial derivative from order two (parabolic) to order one (first order hyperbolic) demonstrates the effect of the spatial fractional order ?? with 1 = ?? = 2. All of the example SFPDEs are one dimensional in Cartesian coordinates. Extensions to higher dimensions and other coordinate systems, in principle, follow from the examples in this second volume. The examples start with a statement of the integer PDEs that are then extended to SFPDEs. The format of each chapter is the same as in the first volume. The R routines can be downloaded and executed on a modest computer (R is readily available from the Internet).
商品描述(中文翻譯)
偏微分方程（PDEs）是科學和工程中最廣泛使用的數學形式之一。PDEs 可以對（1）初始值變數（通常是時間）和（2）邊界值變數（通常是空間變數）進行偏導數。因此，可以考慮兩種分數型 PDE：（1）時間上的分數型（TFPDEs），以及（2）空間上的分數型（SFPDEs）。這兩卷書的重點是 SFPDEs 的發展和應用，討論內容分為：• 第1卷：R 語言中的算法與計算機編碼介紹 • 第2卷：來自經典整數 PDE 的應用。已提出多種空間分數導數的定義。我們專注於 Caputo 導數，並偶爾提及 Riemann-Liouville 導數。在第二卷中，重點是 SFPDEs 的應用，主要通過將經典整數 PDE 擴展到 SFPDEs 來發展。示例應用包括：• 帶有 Dirichlet、Neumann 和 Robin 邊界條件的分數擴散方程 • Fisher-Kolmogorov SFPDE • Burgers SFPDE • Fokker-Planck SFPDE • Burgers-Huxley SFPDE • Fitzhugh-Nagumo SFPDE。這些 SFPDEs 被選擇是因為它們在時間上是整數一階，在空間上是整數二階。空間導數從二階（拋物型）變化到一階（第一階雙曲型）展示了空間分數階 ?? 的影響，其中 1 = ?? = 2。所有示例 SFPDEs 在笛卡爾坐標系中都是一維的。原則上，對於更高維度和其他坐標系的擴展可以從第二卷中的示例中推導出來。這些示例從整數 PDE 的陳述開始，然後擴展到 SFPDEs。每章的格式與第一卷相同。R 程序可以下載並在普通計算機上執行（R 可以輕鬆從互聯網獲得）。
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