NationMaster - Encyclopedia: 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. It corresponds to imposing both a Dirichlet and a Neumann boundary condition. Euclid, detail from The School of Athens by Raphael. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an differential equation that contains functions of only one independent variable, and derivatives in that variable. ... 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. ... The word Boundary has a variety of meanings. ... Domain has several meanings: // General some kind of territory, such as (for example) a demesne or a realm synonymous with a metaphorical field, e. ... 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 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. ... 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. ...

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., In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ...

$y(a)=alpha ,$

and

$y'(a)=beta .$

where $a$ 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.

$u_x = {part u over part x}$

$u_{xy} = {part^2 u over part y, part x}$

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

$psi_{xx} + psi_{yy}= psi(x,y)$

We have a two dimensional domain whose boundary is a boundary line, which in turn can be described by the following parametric equations Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, parametric equations are a bit like functions: they allow someone to fill in some variables, called parameters or independent variables, with any values they wish. ...

$x=&# 0;(s)$

$y=eta (s)$

hence, in a simmilar 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

$psi (s)$

and

$frac{dpsi}{dn}(s)=mathbf{n}cdotnablapsi$

are specified at each point on the boundary of the domain of the given partial differential equation (PDE), where $nablapsi(s) ,$ is the gradient of the function. It is sometimes said that Cauchy boundary conditions are a weighted average of imposing Dirichlet boundary conditions and Neumann boundary conditions. This should not be confused with statistical objects such as the weighted mean, 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 refers to "An average that takes into account the proportional relevance of each component, rather than treating each component equally." (http://www.investorwords.com/5854/weighted_average.html), hence it means that while posing the problem of solving a given PDE, the mathematician, physicists, engineer or whoever is trying to solve it, should try to bear in mind what information there is available for the well-posedness of the problem and its subsequent successful solution. Domain has several meanings: // General some kind of territory, such as (for example) a demesne or a realm synonymous with a metaphorical field, e. ... 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. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... Look up Function on Wiktionary, the free dictionary In general (not in the mathematical but in the engineering sense), a function is a goal-oriented property of an entity (according to the Adam Maria Gadomskis TOGA meta-theory, 1993). ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... Neumann (German meaning new man) may refer to: Alfred Neumann, a writer Erich Neumann (politician) Erich Neumann (psychologist) Günter Neumann, a German cabaretist Hartwig Neuamnn, author of Stadt und Festung Jülich auf bildlichen Darstellungen Balthasar Neumann, Johann Balthasar Neumann, Bohemian John Neumann, Saint John Nepomucene Neumann, Bohemian John von Neumann... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted geometric mean is calculated as Note that if all the weights are equal, the weighted geometric mean is the same as the geometric mean. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted harmonic mean is calculated as Note that if all the weights are equal, the weighted harmonic mean is the same as the harmonic mean. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... The mathematical term well-posed problem stems from a definition given by Hadamard. ...

Since the parameter $s$ is usually time, Cauchy conditions cane also be called intial 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 than having Robin or impedance boundary condition, a mixture of Dirichlet and Neumann boundary conditions are given by

$alpha (s)psi (s)+ beta (s) frac{dpsi }{dn}(s)=f(s)$

here $alpha (s)$ , $beta (s)$ , and $f(s)$ 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 fullfill a condition within the same equation for the boundary condition.

Example

Let us define the heat equation in two spatial dimensions as follows The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

$u_t = k Delta u$

where $k$ is a material-specific constant called thermal conductivity. In physics, thermal conductivity, Î» or k, is the intensive property of a material which relates its ability to conduct heat. ...

and suppose that such equation is applied over the region $G$ , which is the upper semidisk centered at the origin of radius $a$ . 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

$u=0 forall (x,y) in r=a, 0leq theta leq pi$

and

$u_y = 0, y = 0$

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

$u(x,y,t)= phi (x,y) psi (t)$

applying such product to the original equation we obtain

$phi (x,y) psi ' (t)= k phi '' (x,y) psi (t)$

whence

$frac{psi '(t)}{k psi (t)} = frac{phi '' (x,y)}{phi (x)}$

Since the left hand side (l.h.s.) depends only on $t$ , and the right hand side (r.h.s) depends only on $(x,y)$ , we conclude that both should be equal to the same constant

$frac {psi '(t)}{k psi (t)}= - lambda = frac {phi '' (x,y)}{phi (x)}$

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

$phi_{xx}+phi_{yy}+lambda phi (x,y)=0$

and a second equation in the $t$ variable,

$psi '(t) +lambda k psi (t)=0$

Once we impose the boundary conditions, the solution of the temporal ODE is Ode is a form of stately and elaborate lyrical verse. ...

$psi (t) =A e^{-lambda k t}$

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 $phi (x,y) = X(x)Y(y)$ into the PDE and dividing by $X(x) Y(y)$ from which we obtain (after reorganizing terms) Wikipedia does not yet have an article with this exact name. ...

$frac {Y''}{Y}+lambda =-frac {X''}{X}$

since the l.h.s depends only on y and r.h.s only depends on $x$ , both sides must equal a constant, say $mu$ ,

$frac {Y''}{Y}+ lambda =- frac {X''}{X} = mu$

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

External links

References

Cooper, Jeffery M. "Introduction to Partial Differential Equations with MATLAB". ISBN 0-8176-3967-5

Categories: Mathematical analysis stubs | Partial differential equations

Results from FactBites:

Augustin Louis Cauchy - Wikipedia, the free encyclopedia (726 words)
	In 1833 the deposed king Charles X of France summoned Cauchy to be tutor to his grandson, the duke of Bordeaux, an appointment which enabled Cauchy to travel and thereby become acquainted with the favourable impression which his investigations had made.
	Returning to Paris in 1838, Cauchy refused a proffered chair at the Collège de France, but in 1848, the oath having been suspended, he resumed his post at the École Polytechnique, and when the oath was reinstituted after the coup d'état of 1851, Cauchy and François Arago were exempted from it.
	Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849; and Eugène François Cauchy (1802–1877), a publicist who also wrote several mathematical works.

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