mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the field of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

s, a boundary value problem is a

together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the

wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...

, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

s is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to

Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...

); the solution was given by the

Dirichlet's principle In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. Formal statement Dirichlet's principle states that, if the funct ...

Explanation

Boundary value problems are similar to

initial value problem In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...

s. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value). A boundary value is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component. For example, if the independent variable is time over the domain ,1 a boundary value problem would specify values for

y(t)

at both

t=0

and

t=1

, whereas an initial value problem would specify a value of

y(t)

and

y'(t)

at time

t=0

. Finding the temperature at all points of an iron bar with one end kept at

absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...

and the other end at the freezing point of water would be a boundary value problem. If the problem is dependent on both space and time, one could specify the value of the problem at a given point for all time or at a given time for all space. Concretely, an example of a boundary value problem (in one spatial dimension) is :

y''(x)+y(x)=0

to be solved for the unknown function

y(x)

with the boundary conditions :

y(0)=0, \ y(\pi/2)=2.

Without the boundary conditions, the general solution to this equation is :

y(x) = A \sin(x) + B \cos(x).

From the boundary condition

y(0)=0

one obtains :

0 = A \cdot 0 + B \cdot 1

which implies that

B=0.

From the boundary condition

y(\pi/2)=2

one finds :

2 = A \cdot 1

and so

A=2.

One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is :

y(x)=2\sin(x).

Types of boundary value problems

Boundary value conditions

A boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space. A boundary condition which specifies the value of the

normal derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...

of the function is a Neumann boundary condition, or second-type boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known. If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition.

Examples

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

specified by the boundary conditions.

Differential operators

Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which imp ...

, one discusses elliptic boundary value problems. For a hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

and various nonlinear types.

Applications

Electromagnetic potential

electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...

, a common problem is to find a function which describes the

electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...

of a given region. If the region does not contain charge, the potential must be a solution to

(a so-called harmonic function). The boundary conditions in this case are the

Interface conditions for electromagnetic fields Interface conditions describe the behaviour of electromagnetic fields; electric field, electric displacement field, and the magnetic field at the interface of two materials. The differential forms of these equations require that there is always an ...

. If there is no

current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...

in the region, it is also possible to define a magnetic scalar potential using a similar procedure.

Notes

References

* A. D. Polyanin and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)'', Chapman & Hall/CRC Press, Boca Raton, 2003. . * A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. .

External links

* *
Linear Partial Differential Equations: Exact Solutions and Boundary Value Problems
at EqWorld: The World of Mathematical Equations. * {{Authority control Ordinary differential equations Mathematical problems