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- Cauchy_boundary_condition abstract "In mathematics, a Cauchy boundary condition /koʊˈʃiː/ augments an ordinary differential equation or a partial differential equation to completely determine the solution. Thereby, both function value and normal derivative are specified on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. 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, in order to have a particular solution, one must 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 equationshence, in a similar manner as for second-order ordinary differential equations, we now need to know the value of the function at the boundary, as well as its normal derivative, in order to solve the partial differential equation, that is to say, bothand are specified at each point on the boundary of the domain of the given partial differential equation (PDE), where 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 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 a Robin or impedance boundary condition. A mixture of Dirichlet and Neumann boundary conditions is 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.".
- Cauchy_boundary_condition wikiPageID "1725025".
- Cauchy_boundary_condition wikiPageRevisionID "603787051".
- Cauchy_boundary_condition hasPhotoCollection Cauchy_boundary_condition.
- Cauchy_boundary_condition title "Cauchy boundary conditions".
- Cauchy_boundary_condition urlname "CauchyBoundaryConditions".
- Cauchy_boundary_condition subject Category:Boundary_conditions.
- Cauchy_boundary_condition type Abstraction100002137.
- Cauchy_boundary_condition type BoundaryCondition106755776.
- Cauchy_boundary_condition type BoundaryConditions.
- Cauchy_boundary_condition type Communication100033020.
- Cauchy_boundary_condition type Condition106755568.
- Cauchy_boundary_condition type Message106598915.
- Cauchy_boundary_condition type Postulate106753299.
- Cauchy_boundary_condition type Premise106753800.
- Cauchy_boundary_condition type Proposition106750804.
- Cauchy_boundary_condition type Statement106722453.
- Cauchy_boundary_condition comment "In mathematics, a Cauchy boundary condition /koʊˈʃiː/ augments an ordinary differential equation or a partial differential equation to completely determine the solution. Thereby, both function value and normal derivative are specified on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition.".
- Cauchy_boundary_condition label "Cauchy boundary condition".
- Cauchy_boundary_condition label "Condición de frontera de Cauchy".
- Cauchy_boundary_condition label "コーシー境界条件".
- Cauchy_boundary_condition sameAs Condición_de_frontera_de_Cauchy.
- Cauchy_boundary_condition sameAs Condizioni_al_contorno_di_Cauchy.
- Cauchy_boundary_condition sameAs コーシー境界条件.
- Cauchy_boundary_condition sameAs m.05rb_g.
- Cauchy_boundary_condition sameAs Q3278575.
- Cauchy_boundary_condition sameAs Q3278575.
- Cauchy_boundary_condition sameAs Cauchy_boundary_condition.
- Cauchy_boundary_condition wasDerivedFrom Cauchy_boundary_condition?oldid=603787051.
- Cauchy_boundary_condition isPrimaryTopicOf Cauchy_boundary_condition.