boundary condition

Mathematical terminology

Collection

zero Useful+1

zero

Boundary conditions refer to the boundary conditions Solution area The variation law of the variable or its derivative solved on the boundary with time and place. The boundary condition is governing equation There is the premise of determining the solution. For any problem, the boundary conditions need to be given. The treatment of boundary conditions directly affects the accuracy of calculation results. And the solution differential equation To have a definite solution, conditions must be introduced, and these additional conditions are called Definite solution condition 。

Chinese name: boundary condition

Foreign name: boundary condition
Type: Mathematical terminology

catalog

one brief introduction
two classification

brief introduction

Announce

edit

If the equation requires Unknown quantity Y (x) and its derivative y ′ (x) in independent variable The same point of x=x zero Take the given value, that is, y (x zero )=y zero ，y′(x zero )= y zero ', then this condition is called the initial condition, and the problem composed of equations and initial conditions is called Initial value problem ；

However, in many practical problems, it is often required that the solution of the differential equation meets certain conditions at the endpoint of a ≤ x ≤ b in a given interval. For example, if y (a)=A, y (b)=B, then（ Boundary point ）The condition of the value of is called boundary condition, which is composed of differential equation and boundary condition mathematical model It is called boundary value problem.

classification

Announce

edit

The boundary condition in the boundary value problem can be expressed as follows at the end: Ay+By '=C

① If B=0, A ≠ 0: The first kind of boundary value condition , also known as Dirichlet boundary condition 。

It gives the value of the unknown function on the boundary, which directly describes the physical quantity on the boundary of the physical system, such as the distance between the two ends of the vibrating string and the balance position;

② If B ≠ 0, A=0: Boundary value condition of the second kind , also known as the Neumann boundary condition.

It gives the directional derivative of the normal of the unknown function outside the boundary, and describes the derivative of the physical quantity on the boundary of the physical system perpendicular to the boundary, such as the heat flow at the end of the heat conducting thin rod;

③ If A ≠ 0, B ≠ 0: Boundary condition of the third kind , also known as Robin condition.

It describes the linear combination of physical quantities on the boundary of the physical system and the derivatives of the vertical boundary. For example, the free cooling at the end of a thin rod, the temperature, and the heat flow are uncertain, but the relationship between the two is determined, so the boundary value conditions formed by the linear combination of the two can be listed.

Additional points initial condition ：

The initial condition refers to the initial state of the process, that is, the unknown function and its order to time partial derivative The value of t=0 at the initial time Finite element Many initial conditions should be given in advance. different Field equation Corresponding to different initial conditions.

In short, in order to determine the universal definition Solution of the equation , you must provide sufficient initial conditions and boundary conditions!

Neumann boundary condition

In mathematics, Neumann Neumann boundary condition ordinary differential equation or partial differential equation Of“ Boundary condition of the second kind ”。 The Neumann boundary condition specifies the differentiation of the solution of the differential equation at the boundary.

In the case of ordinary differential equations, such as

In the interval [0,1], the Neumann boundary condition has the following form: