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The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
Domain decomposition methods. In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains.
To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image). This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner.
For a domain having a sufficiently smooth boundary , the general solution to the Dirichlet problem is given by = (,),where (,) is the Green's function for the partial differential equation, and
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.
In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into non-overlapping subdomains, and the unknowns in the interiors of the subdomains are eliminated.
An initial value problem is a differential equation ′ = (, ()) with : where is an open set of , together with a point in the domain of (,),called the initial condition.. A solution to an initial value problem is a function that is a solution to the differential equation and satisfies