Cauchy boundary condition

In mathematics, a Cauchy (pronounced "koe-she") 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the It corresponds to imposing both a Dirichlet and a Neumann boundary condition. In Mathematics, the Dirichlet (or first type) boundary condition is a type of Boundary condition, named after Johann Peter Gustav Lejeune In Mathematics, the Neumann (or second type) boundary condition is a type of Boundary condition, named after Carl Neumann. It is named after the prolific 19th century French mathematical analyst Augustin Louis Cauchy.

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. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its e. ,

$y(a)=\alpha \ ,$

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

$x=\xi (s) \$

$y=\eta (s) \$

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

$\psi (s) \$

and

$\frac{d\psi}{dn}(s)=\mathbf{n}\cdot\nabla\psi \$

are specified at each point on the boundary of the domain of the given partial differential equation (PDE), where $\nabla\psi(s) \,$ is the gradient of the function. In Mathematics, parametric equations are a method of defining a curve In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function It is sometimes said that Cauchy boundary conditions are a weighted average of imposing Dirichlet boundary conditions and Neumann boundary conditions. The weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints Neumann ( German for "new man" and one of the 20 most common German surnames may refer to Alfred Neumann, German playwright In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints 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. The weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average In Statistics, given a set of data X = { x 1 x 2. x n } and corresponding 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. The weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average The mathematical term well-posed problem stems from a definition given by Hadamard.

Since the parameter $s \$ 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 or impedance boundary condition, a mixture of Dirichlet and Neumann boundary conditions are given by

$\alpha (s)\psi (s)+ \beta (s) \frac{d\psi }{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 Mathematics, the Robin (or third type) boundary condition is a type of Boundary condition, named after Victor Gustave Robin (1855-1897 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 in two spatial dimensions as follows

$u_t = k \nabla^2 u \$

where $k \$ is a material-specific constant called thermal conductivity. The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time In Physics, thermal conductivity, k is the property of a material that indicates 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, 0\leq \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

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

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

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

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

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

where A is a constant which could be defined upon the initial conditions. Ode (from the Ancient Greek) is a form of stately and elaborate lyrical verse. 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)

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

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

External links

References

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

http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-303Fall-2004/88A7832B-A69E-48B9-A566-2F3B810164F0/0/pde3d.pdf

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