In mathematics, in the field of differential equations, a boundary value
problem is a differential equation 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, 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. 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 equations 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; the solution was given by the
Dirichlet's principle. Explanation
Boundary value problems are similar to initial value problems. A boundary value
problem has conditions specified at the extremes 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.
For example, if the independent variable is time over the domain [0,1], a
boundary value problem would specify values for at both and , whereas an
initial value problem would specify a value of and at time .
Finding the temperature at all points of an iron bar with one end kept at
absolute zero 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 the data or at a given time for all space.
Concretely, an example of a boundary value is the problem
to be solved for the unknown function with the boundary conditions
Without the boundary conditions, the general solution to this equation is
From the boundary condition one obtains which implies that From the boundary
condition one finds and so One sees that imposing boundary
conditions allowed one to determine a unique solution, which in this case is
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 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, , constants and specified by the boundary conditions,
and a known scalar function 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, one discusses
elliptic boundary value problems. For an hyperbolic operator, one discusses
hyperbolic boundary value problems. These categories are further subdivided
into linear and various nonlinear types. Applications
= Electromagnetic potential= In electrostatics, a common problem is
to find a function which describes the electric potential of a given region. If
the region does not contain charge, the potential must be a solution to
Laplace's equation.(a so-called harmonic function) The boundary conditions in
this case are the Interface conditions for electromagnetic fields. If there is
no current density in the region, it is also possible to define a magnetic
scalar potential using a similar procedure.
See also References
A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary
Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN
1-58488-297-2. A. D. Polyanin, Handbook of Linear
Partial Differential Equations for Engineers and Scientists, Chapman &
Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9.
External links Hazewinkel, Michiel, ed., "Boundary
value problems in potential theory", Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4 Hazewinkel, Michiel, ed., "Boundary
value problem, complex-variable methods", Encyclopedia of Mathematics,
Springer, ISBN 978-1-55608-010-4 Linear Partial Differential Equations:
Exact Solutions and Boundary Value Problems at EqWorld: The World of
Mathematical Equations. Boundary value problem at Scholarpedia.