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The problem has a solution; The solution is unique; The solution's behavior changes continuously with the initial conditions; Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions. These might be regarded as 'natural' problems in that there are ...
The problem has been well-known ever since it was originally posed by Bernhard Riemann in 1860. The Clay Institute's exposition of the problem was given by Enrico Bombieri . [ 18 ]
A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique ...
In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. [ citation needed ] More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface .
Thus we require some assumption on the matrix to create a well-posed problem, such as assuming it has maximal determinant, is positive definite, or is low-rank. [ 1 ] [ 2 ] For example, one may assume the matrix has low-rank structure, and then seek to find the lowest rank matrix or, if the rank of the completed matrix is known, a matrix of ...
The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is a well-posed problem in the sense of Hadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution .
In addition to making humorous posts about daily mom issues, including how difficult it is to get a family photo where all children look at the camera, the 39-year-old also keeps it real regarding ...
"Well-posed problem: A problem that has a unique solution that depends continuously on the initial data." "Ill-posed problem: [MATH] A problem which may have more than one solution, or in which the solutions depend discontinuously on the initial data. Also known as improperly posed problem."