Mathematical Problems

Mathematical problems in partial differential equations begin with a relation between an unknown function and its partial derivatives. But in most applications, the equation alone is not enough. We also need extra information to select the function that fits the given situation. For this reason, a mathematical problem in partial differential equations usually contains two parts. The first part is the PDE. The second part is a set of suitable conditions.

A mathematical problem in a partial differential equation is the problem of finding an unknown function which satisfies a given partial differential equation and the given supplementary conditions.

Here, the word supplementary means additional. These conditions are added to the equation so that the required function becomes properly determined.

Supplementary conditions are extra conditions attached to a partial differential equation. They may be initial conditions, boundary conditions, or both. Sometimes they may also involve derivatives of the unknown function. Initial conditions usually describe the value of the unknown function at an initial time, such as

t=0

t=t_0

. Boundary conditions describe the value of the unknown function, or sometimes its derivative, on the boundary of the spatial region.

The interactive classifier below helps distinguish different kinds of supplementary conditions. Notice that an initial condition usually refers to a starting time, while a boundary condition refers to the edge of the spatial region. A condition involving a derivative may also appear as supplementary information.

The following example shows how a PDE and its supplementary conditions together form a complete mathematical problem.

Consider the heat equation

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

Here

u=u(x,t)

is the unknown function. The variable

x

represents position in a rod, and

t

represents time. Suppose the following conditions are also given:

\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 first condition,

u(x,0)=\sin x

, gives the value of the unknown function at the initial time

t=0

. So it is an initial condition. The second and third conditions give the values of the unknown function at the two boundary points

x=0

and

x=l

. So they are boundary conditions. Thus the full problem consists of one partial differential equation and three supplementary conditions.

The preview below represents the rod as the interval from

x=0

x=l

. The curve shows the initial temperature

u(x,0)=\sin x

, while the endpoint markers represent the boundary conditions. As the rod length changes, observe how the initial profile and the right boundary endpoint are related.

The calculator below checks whether the initial condition and the right boundary condition agree at the corner point

(l,0)

. Since the initial condition gives

u(l,0)=\sin l

and the boundary condition gives

u(l,0)=0

, compatibility at that point requires

\sin l=0

In this example, the PDE describes heat flow in a rod of length

l

. The unknown function

u(x,t)

may be understood as the temperature at the point

x

and at the time

t

. The initial condition tells us how the temperature is distributed in the rod at the starting time. The boundary conditions tell us what happens at the two ends of the rod.

A partial differential equation may have many possible solutions. Supplementary conditions help us choose the solution that matches the actual problem. For example, the heat equation describes a general law of heat flow. But the temperature in a particular rod also depends on the starting temperature and on what happens at the ends of the rod. So the PDE gives the rule, and the supplementary conditions give the required data for the particular situation.

A problem that contains both an initial condition and boundary conditions is called an Initial Boundary-Value Problem. This idea is closely related to Initial Conditions and Boundary Conditions. Later, when we study Well-Posed Problems, we shall see that a good mathematical problem should have a solution, should not have more than one solution, and should depend continuously on the given data

BMLabs Mathematics

BMLabs Mathematics

Mathematical Problems

Mathematical Problems

Supplementary Condition Classifier

Heat Rod Condition Preview

Condition Compatibility Checker

Coordinates and Locus

Invariants under Orthogonal Transformation