這篇文章主要探討曲線上偏微分方程數值解的一個新方法，其計算域將會是曲 線周圍的管狀區域，並且使用歐氏空間的等距分割點來離散化，Neumann 邊界條 件則是使用插值法來處理。即使是未參數化的曲線，也能輕易適用此方法。我們將 使用此方法來模擬天線模型，此模型結合了 Telegrapher 方程式和 Maxwell 方程 式，並且描述電線中的電流如何和空間中的電磁場產生交互作用。其中，我們使 用時域有限差分法，並搭配 Perfectly Matched Layer 邊界條件來解 Maxwell 方程 式，而 Telegrapher 方程式則是使用我們所提出的新方法來處理。文內亦提出一些 應用此方法的範例。

In this thesis, we propose a new method using volumetric extension to solve PDEs on a curve. The computational domain is a narrow tubular region surrounding the curve. Our method uses Cartesian grids on the narrow tube and the Neumann boundary condition is taken care by interpolation treatment. Therefore it is easy to implement, even for complicated curves without given parametrization. We apply our method for thin-wire model equations. The model combines the telegrapher's equations with the Maxwell's equations, and it describes how an electrical wire reacts with nearby electric and magnetic fields. The Maxwell's equations are solved by Finite-Difference Time-Domain (FD-TD) method with perfectly matched layer (PML) boundary condition and the telegrapher's equations are solved by our approach. Some numerical examples are presented.

1 Introduction 1
2 FD-TD method 3
2.1 Algorithm................................... 3 2.2 Boundaryconditions ............................. 5
3 Volumetric extension of PDEs 6
3.1 Outline..................................... 6 3.2 Maintheoremsandproofs .......................... 7
4 Numerical examples 11
4.1 Poissonequationonacurve ......................... 11 4.2 Heatequationonacurve........................... 13 4.3 Telegrapher’sequationsontheunitcircle . . . . . . . . . . . . . . . . . . 14
5 Coupling between the wire and the space 15

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