# 數學~微分方程式跟積分方程式之應用比較

### 2 個解答

• 1 0 年前
最佳解答

1. 有積分方程式. 原則上凡方程式包括未知函數的積分即可歸類成積分方程式.

2. 積分方程式之分類:首先亦有線性與非線性二種. 線性裡依積分性質分二大類別: definite integral --> Fredholm equation; indefinite integral --> Volterra equations; 每一種又有first kind &second kind 二型{ 也有人分三型的}.

examples: 1. u(x)-int[t=0 to x] { e^(st) u(t)} dt = x+1 is a Volterra integral equation of the second kind for the unknown u=u(x); 2, u(x)-int[t=0 to 1] { e^(st) u(t)} dt = x+1 is a Fredholm integral equation of the second kind for the unknown u=u(x); 3. int[t=0 to 1] { e^(st) u(t)} dt = x+1 is a Fredholm integral equation of the first kind for the unknown u=u(x). [ without the linear term of u(x)].

3. 微分方程式和積分方程式往往是一體的兩面.

e. g. An initial value problem [DE + initial conditions]can be written as a Volterra integral equation, and vice versa; A boundary value problem [ DE + boundary conditions] can be written as a Fredholm integral equation, and vice versa.

In fact the famous Picard theorem [proving the existence and uniqueness of the solution to initial value problem for a first order ode ] is derived using its equivalent integral equation.

4. 探討積分方程式之解存在與否需要相當的數學基礎[高等微積分與泛函分析]. 故積分方程只會有可能在大一點的數學系研究所裡開成課. 如若大大有興趣一探究竟, 我建議這一本聖經: Integral Equations by Harry Hochstadt.

有了初步的認識才好談其應用.

• 德馨
Lv 6
1 0 年前

但是屬於研究所的課程