線性橢圓偏微分方程

Abstract

研究線性橢圓偏微分方程(線性橢圓PDEs) 。首先, 給一些實用的例子, 同時將二階線性偏微分方程式作一分類。接下來, 運用幾個古典方法解線性橢圓偏微分方程,並且將該方程式的解以各種形式表示。

當我們運用傅立葉轉換解整個或半平面的偏微分方程時, 需要利用逆傅立葉轉換導出該偏微分方程的解, 此時被積分函數中常出現平方根的形式, 在複數平面上它是多值函數。為了讓逆傅立葉轉換導出的解是正確的, 我們結合複數平面上的黎曼曲面,藉由適當的代數建構出平方根在該曲面上是單值, 並且完成逆轉換的解析解與數值解。最後藉由例子來說明整個計劃。

We study the linear elliptic partial differential equations ( linear elliptic PDEs ). First, we give some practical examples and show that they are governed by such type of the equations. Next, we apply several classical methods to solve the linear elliptic PDEs with the solutions being expressed in various forms. We then identify those solutions.

When we apply Fourier transformations to the whole- and half-line PDEs, it is necessary to perform the inverse Fourier transformations to derive the PDE solutions, and it is quite often that those integrals involve the square root operator which is multi-valued in the complex plane. In order to perform the inverse transformations correctly, we develop the Riemann surfaces from the complex plane with the proper algebraic structures to assure that the square root is now a single-valued function on the surfaces, and we are able to accomplish the inverse transformations analytically and numerically. Some examples are given to illustrate the entire scheme.