偏微分方程中的Fourier变换及其应用

The Theory of Fourier Transform And Applied In Partlal Differential Equations

Ｆｏｕｒｉｅｒ变换在偏微分方程理论中占有相当重要的地位。本文通过经典的Ｆｏｕｒｉｅｒ变换求解一维、二雏热传导方程Ｃａｕｃｈｙ问题的方法，推广Ｆｏｕｒｉｅｒ变换到速降函数空间ψ（Ｒ￣ｎ）和ψ（Ｒ￣ｎ）上的线性连续泛函ψ＇（Ｒ￣ｎ）广义函数空间上，求出了Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程、热传导方程、常系数椭圆算子方程以及形如的方程的Ｃａｕｃｈｙ问题的基本解。最后引入了Ｆｏｕｒｉｅｒ变换的其它几种形式，并用三角函数形式的Ｆｏｕｒｉｅｒ变换，求解了两个物理实际问题。

Fourier transform plays a very important role in the theory of partial differential equa-tions.In this article,first,we solve the problems of one-dimensional and two-dimensionalheat conduction equations with only initial conditions(Cauchy problem)from the general de-fination of Fourier transform. Secondly,we give the defination of Fourier transform on ψ (R￣n)andψ'(R￣n).find the basic solutions of Cauchy-Riemann equation、heat conduction equa-tion、the elliptic operator with contstant coefficents and the equations form as with only initial conditions. Finally,we use the other forms of Fourier transform and solve two physical problems from the Fourier sine and cosine integrals transforms.