Shows a region where a differential equation is valid and the associated boundary values

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equations which also satisfies the boundary conditions. Image File history File links Size of this preview: 373 × 158 pixelsFull resolution (373 × 158 pixel, file size: 6 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Boundary value problem Elliptic boundary value... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

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. The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ... Various normal modes in a 1D-lattice. ... In mathematics and its applications, a Sturm-Liouville problem, named after Charles Francois Sturm (1803-1855) and Joseph Liouville (1809-1882), is a second-order linear differential equation of the form (1) often together with specified boundary values of y and dy/dx. ... In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation 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. The mathematical term well-posed problem stems from a definition given by Hadamard. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet principle. In mathematics, Dirichlet problems are a class of partial differential equation (PDE) problems which ask you to solve for the values of a function in a region given the value of the function on the boundary of that region. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ... In mathematics, Dirichlets principle in potential theory states that the harmonic function on a domain with boundary condition on can be obtained as the minimizer of the Dirichlet integral amongst all functions such that on , provided only that there exists one such function making the Dirichlet integral finite. ...

Initial value problem

Main article: Initial value problem

A more mathematical way to picture the difference between an initial value problem and a boundary value problem is that an initial value problem has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower boundary of the domain, thus the term "initial" value). On the other hand, a boundary value problem has conditions specified at the extremes of the independent variable. For example, if the independent variable is time over the domain [0,1], an initial value problem would specify a value of y(t) and y'(t) at time t = 0, while a boundary value problem would specify values for y(t) at both t = 0 and t = 1. In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ...

If the problem is dependent on both space and time, then instead of specifying the value of the problem at a given point for all time the data could be given at a given time for all space. For example, the temperature 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. Whereas in a the middle of a still pond if somebody taps the water with a known force that would create a ripple and give us an initial condition. Absolute zero is the lowest possible temperature where nothing could be colder, and no heat energy remains in a substance. ...

Types of boundary value problems

The boundary value problem for an idealised 2D rod

If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example if one end of an iron rod had a heater at one end then energy would be added at a constant rate but the actual temperature would not be known. Image File history File links Size of this preview: 529 × 205 pixelsFull resolution (529 × 205 pixel, file size: 6 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Boundary value problem ... :For other senses of this word, see dimension (disambiguation). ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector intuitively represents the rate of change of the function in the direction of that vector. ... In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain. ...

If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For example if one end of an iron rod had one end held at absolute zero then the value of the problem would be known at that point in space. In mathematics, a Dirichlet boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values a solution is to take on the boundary of the domain. ...

If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition. In mathematics, a Cauchy boundary condition imposed on an ordinary differential equation or a partial differential equation specifies both the values a solution of a differential equation is to take on the boundary of the domain and the normal derivative at the boundary. ...

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. In mathematics, an elliptic operator is one of the major types of differential operator P. It can also be defined on spaces of complex-valued functions, or some more general function-like objects. ... Shows a region where a differential equation is valid and the associated boundary values 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. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ...

Example

Consider the ordinary differential equation We will use k to denote the square root of the absolute value of . ...

to be solved for the unknown function y(x). Impose the boundary conditions

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

From the boundary condition y(0) = 0 one obtains

which implies that B = 0. From the boundary condition y(π / 2) = 2 one finds

and so A = 2. One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case is

See also

• Related mathematics:
  • initial value problem
  • differential equations
  • Green's functions
• Physical applications:
  • waves
  • normal modes
  • electrostatics
  • Laplace equation
  • potential theory
• numerical algorithm: shooting method

In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ... This article is about waves in the most general scientific sense. ... Normal modes in an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency (called normal frequencies or allowed frequencies). ... Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ... Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ... Potential theory may be defined as the study of harmonic functions. ... In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to the solution of an initial value problem. ...

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. 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

• Linear Partial Differential Equations: Exact Solutions and Boundary Value Problems at EqWorld: The World of Mathematical Equations.