線性雙曲型偏微分方程之研究

Abstract

本論文主要研究線性的雙曲型偏微分方程(線性雙曲PDEs)。首先，我們舉給幾個屬於此類型的實際例子。再來，使用幾種典型的方法來解線性雙曲PDEs。同時並以不同形式來表示解，並且確定解的一致性。 當我們對PDEs使用積分轉換時(對於變數是整條實數線使用Fourier轉換；變數是半射線使用Laplace轉換)，再藉由逆積分轉換(inversion Fouier transform or inversion Laplce transform)來得到PDEs的解是必要的。但是執行逆積分轉換時，經常那些被積分會出現平方根。然而，平方根在複數平面上是多值的。為了能正確地進行逆轉換，我們利用適當的代數分析來建構多值函數的黎曼曲面，使其變成單值函數，以致於我們能正確地在分析上和數值上完成逆積分轉換。最後由一些例子說明整個架構。

We study the linear hyperbolic partial differential equations (linear hyperbolic 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 hyperbolic PDEs with the solutions being expressed in various forms. We then identify those solutions.

When we apply Fourier and Laplace transformations to the whole- and half-line PDEs,it is necessary to perform the inverse Fourier and Laplace 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.