![Boundary value problem-en](https://upload.wikimedia.org/wikipedia/commons/2/2c/Boundary_value_problem-en.svg)
In
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, an ...
s, a boundary value problem is a
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, an ...
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 s ...
, such as the determination of
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s, 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
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s 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 equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
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
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
, of finding the
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
s (solutions to
Laplace's equation); 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 functi ...
.
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 ...
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
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
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 vibration ...
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
:
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
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
, 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
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appl ...
, 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 In mathematics, a Cauchy () boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Ca ...
.
Examples
Summary of boundary conditions for the unknown function,
, constants
and
specified by the boundary conditions, and known scalar functions
and
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 problem
In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distri ...
s. For a
hyperbolic operator
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can b ...
, one discusses
hyperbolic boundary value problems
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
. 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 r ...
and various nonlinear types.
Applications
Electromagnetic potential
In
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 amber ...
, 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
Laplace's equation (a so-called
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
). 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
Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism analogous to electric potential. It is used to specify the magnetic H-field in cases when there are no free currents, in a manner analogous to using the electric p ...
using a similar procedure.
See also
Related mathematics:
*
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 ...
*
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
*
Stochastic processes and boundary value problems In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. Howe ...
*
Sturm–Liouville theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form:
\frac\!\!\left ,p(x)\frac\right+ q(x)y = -\lambda\, w(x)y,
for given functions p(x), q(x) and w(x), together wit ...
*
Sommerfeld radiation condition In applied mathematics, the Sommerfeld radiation condition is a concept from theory of differential equations and scattering theory used for choosing a particular solution to the Helmholtz equation. It was introduced by Arnold Sommerfeld in 1912 ...
*
Perfect thermal contact condition
Physical applications:
*
Wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
s
*
Normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
*
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 amber ...
*
Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
*
Computation of radiowave attenuation in the atmosphere
The computation of radiowave attenuation in the atmosphere is a series of radio propagation models and methods to estimate the path loss due to attenuation of the signal passing through the atmosphere by the absorption of its different components. ...
*
Black hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
Numerical algorithms:
*
Shooting method
*
Direct multiple shooting method
In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sough ...
*
Walk-on-spheres method
In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations (PDEs). The ...
*
Finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
*
Boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, el ...
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 Problemsat EqWorld: The World of Mathematical Equations.
*
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Ordinary differential equations
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