Initial Boundary-Value Problem

An initial boundary-value problem is a partial differential equation problem with information at the starting time and information on the boundary of the region. Both types of information are needed to describe the full problem. This type of problem is common in heat flow, wave motion, and many physical models. The initial condition tells us how the process starts. The boundary conditions tell us what happens at the edge of the region.

An initial boundary-value problem is a problem in which a partial differential equation is given together with an initial condition and one or more boundary conditions.

The preview below separates an initial boundary-value problem into its main parts. We observe that the partial differential equation gives the rule inside the region, the initial condition gives starting information, and the boundary conditions give information at the edge of the region.

The name is useful because it tells us the structure of the problem. The word initial refers to information given at the starting time, such as

t=0

. The word boundary refers to information given on the boundary of the spatial region, such as the endpoints of an interval. The word value means that the problem asks us to find the unknown function that satisfies all the given information.

Consider a rod of length

l

. Let

u(x,t)

denote the temperature at position

x

and time

t

. A simple heat equation model is

\begin{equation*} u_t-u_{xx}=0,\quad 0<x<l,\quad t>0. \end{equation*}

Now suppose the following conditions are prescribed:

\begin{align*} u(x,0) &= \sin x,\quad 0\leq x\leq l \nonumber \ u(0,t) &= 0,\quad t\geq 0 \nonumber \ u(l,t) &= 0,\quad t\geq 0. \nonumber \end{align*}

The condition

u(x,0)=\sin x

is the initial condition. It gives the temperature distribution in the rod at the starting time

t=0

. The conditions

u(0,t)=0

and

u(l,t)=0

are boundary conditions. They give the temperature at the two ends of the rod for

t\geq 0

. Thus this is an initial boundary-value problem, because it contains one partial differential equation, one initial condition, and two boundary conditions.

The preview below shows the heat conduction problem in the

(x,t)

plane. The interior region represents where the partial differential equation holds. The bottom edge represents the initial condition, and the two vertical edges represent the boundary conditions.

The checker below helps decide whether a given collection of information forms an initial boundary-value problem. Notice that both starting information and boundary information are needed, together with the partial differential equation.

In this example, the variables

x

and

t

are independent variables. The variable

x

describes position in the rod. The variable

t

describes time. The unknown function

u(x,t)

must satisfy the heat equation inside the region

0<x<l

, for time

t>0

. It must also satisfy the prescribed initial and boundary conditions.

An initial condition is connected with time. It usually gives the value of the unknown function when

t=0

t=t_0

. A boundary condition is connected with space. It gives information at the boundary of the spatial region. For the rod

0\leq x\leq l

, the boundary points are

x=0

and

x=l

. So conditions at these two points are boundary conditions.

An initial boundary-value problem joins the physical law with the physical situation. The partial differential equation gives the law. In the heat equation, it describes how heat flows. The initial condition gives the starting state. The boundary conditions describe what happens at the edge of the body. Together, these pieces form the full mathematical problem.

Initial Boundary-Value Problem is closely related to Initial Conditions and Boundary Conditions. It is also an important example of Mathematical Problems in Partial Differential Equations. When we study Well-Posed Problems, we shall ask whether such a problem has a solution, whether the solution is unique, and whether the solution changes only slightly when the data are changed slightly.

BMLabs Mathematics

BMLabs Mathematics

Initial Boundary-Value Problem

Initial Boundary-Value Problem

Problem Structure Preview

Heat Rod Condition Map

Initial Boundary Value Checker

Coordinates and Locus

Invariants under Orthogonal Transformation