Maths | Boundary Conditions

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Summary of boundary conditions for the unknown function, \(y\), constants \(c_0\) and \(c_1\) specified by the boundary conditions, and known scalar functions \(f\) and \(g\) pecified by the boundary conditions.

Name	Form on 1st part of boundary	Form on 2nd part of boundary
Dirichlet	\(y = f\)
Neumann	\(\dfrac{\partial y}{\partial n} = f\)
Robin	\(c_0 y + c_1 \dfrac{\partial y}{\partial n} = f\)
Mixed	\(y = f\)	\(c_0 y + c_1 \dfrac{\partial y}{\partial n} = g\)
Cauchy	both \(y = f\) and \(\dfrac{\partial y}{\partial n} = g\)

• 根據條件的形式，邊值條件分以下三類
  • 第一類邊值條件：也稱為狄利克雷邊界條件，直接描述物理系統邊界上的物理量，例如振動的弦兩端與平衡位置的距離；
  • 第二類邊值條件：也稱為諾伊曼邊界條件，描述物理系統邊界上物理量垂直邊界的導數的情況，例如導熱細杆端點的熱流；
  • 第三類邊值條件：物理系統邊界上物理量與垂直邊界導數的線性組合，例如，細杆端點的自由冷卻，溫度、熱流均不確定，但是二者的關係確定，即可列出二者線性組合而成的邊值條件。

Boundary type	Formal Name	Mathematical form
Type 1	Dirichlet	\(h(x,y,z,t) = constant\)
Type 2	Neumann	\(\dfrac{\partial h}{\partial n} = constant\)
Type 3	Cauchy	\(\dfrac{\partial h}{\partial n} + c h= constant\)

邊值條件也可以根據邊值問題對應的微分算子來分類：若是使用橢圓算子，則問題為橢圓邊值問題；使用雙曲線算子，則問題為雙曲線邊值問題。依微分算子還可以將問題再細分為線性及非線性等。