Cauchy boundary condition - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Cauchy boundary condition (icon) imposed on an ordinary differential equation

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

or a partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

specifies both the values a solution of a differential equation is to take on the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

of the domain

Domain (mathematics)

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

and the normal derivative at the boundary. It corresponds to imposing both a Dirichlet

Dirichlet boundary condition

In mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet who studied under Cauchy and succeeded Gauss at University of Göttingen. When imposed on an ordinary or a partial differential equation, it specifies the values a...

and a Neumann boundary condition

Neumann boundary condition

In mathematics, the Neumann boundary condition is a type of boundary condition, named after Carl Neumann.When imposed on an ordinary or a partial differential equation, it specifies the values that the derivative of a solution is to take on the boundary of the domain.* For an ordinary...

. It is named after the prolific 19th century French mathematical analyst Augustin Louis Cauchy

Augustin Louis Cauchy

Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

.

Cauchy boundary conditions can be understood from the theory of second order, ordinary differential equations, where to have a particular solution one has to specify the value of the function and the value of the derivative at a given initial or boundary point, i.e.,

and

where

is a boundary or initial point.

Cauchy boundary conditions are the generalization of these type of conditions. Let us first recall a simplified form for writing partial derivatives.

and let us now define a simple, second order, partial differential equation:

We have a two dimensional domain whose boundary is a boundary line, which in turn can be described by the following parametric equations

hence, in a similar manner as for second order, ordinary differential equations, we now need to know the value of the function at the boundary, and its normal derivative in order to solve the partial differential equation, that is to say, both

and

are specified at each point on the boundary of the domain of the given partial differential equation

Partial differential equation

(PDE), where

is the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of the function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. It is sometimes said that Cauchy boundary conditions are a weighted average of imposing Dirichlet boundary condition

Dirichlet boundary condition

s and Neumann boundary condition

Neumann boundary condition

s. This should not be confused with statistical objects such as the weighted mean

Weighted mean

The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...

, the weighted geometric mean or the weighted harmonic mean, since no such formulas are used upon imposing Cauchy boundary conditions. Rather, the term weighted average means that while analyzing a given boundary value problem, one should bear in mind all available information for its well-posedness and subsequent successful solution.

Since the parameter

is usually time, Cauchy conditions can also be called initial value conditions or initial value data or simply Cauchy data.

Notice that although Cauchy boundary conditions imply having both Dirichlet and Neumann boundary conditions, this is not the same at all as having Robin

Robin boundary condition

In mathematics, the Robin boundary condition is a type of boundary condition, named after Victor Gustave Robin . When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the...

or impedance boundary condition, a mixture of Dirichlet and Neumann boundary conditions are given by

where

, and

are understood to be given on the boundary (this contrasts to the term mixed boundary conditions, which is generally taken to mean boundary conditions of different types on different subsets of the boundary). In this case the function and its derivative must fulfill a condition within the same equation for the boundary condition.

Example

Let us define the heat equation

Heat equation

The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

in two spatial dimensions as follows

where

is a material-specific constant called thermal conductivity

Thermal conductivity

In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

and suppose that such equation is applied over the region

, which is the upper semidisk of radius

centered at the origin. Suppose that the temperature is held at zero on the curved portion of the boundary, while the straight portion of the boundary is insulated, i.e., we define the Cauchy boundary conditions as

and

We can use separation of variables by considering the function as composed by the product of the spatial and the temporal part

applying such product to the original equation we obtain

whence

Since the left hand side (l.h.s.) depends only on

, and the right hand side (r.h.s) depends only on

, we conclude that both should be equal to the same constant

Thus we are led to two equations: the first in the spatial variables

and a second equation in the

variable,

Once we impose the boundary conditions, the solution of the temporal ODE

ODE

ODE may refer to:* Ohio Department of Education, the state education agency of Ohio* Omicron Delta Epsilon, an international honor society in the field of economics* Open Dynamics Engine, a real-time physics engine...

where A is a constant which could be defined upon the initial conditions.
The spatial part can be solved again by separation of variables, substituting

into the PDE

Partial differential equation

and dividing by

from which we obtain (after reorganizing terms)

since the l.h.s depends only on y and r.h.s only depends on

, both sides must equal a constant, say

so we obtain a pair of ODE's upon which we can impose the boundary conditions that we defined

Example

See also