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The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which boundary conditions are satisfied by combining a solution not restricted by the boundary conditions with its possibly weighted mirror image. Generally, original singularities are inside the domain of interest but the function is ...
The Kansa method is a computer method used to solve partial differential equations. Its main advantage is it is very easy to understand and program on a computer. It is much less complicated than the finite element method. Another advantage is it works well on multi variable problems.
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.. The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x 2 − 3x + 2 = 0.
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
Let be a domain (an open and connected set) in .Let be the Laplace operator, let be a bounded function on the boundary, and consider the problem: {() =, = (),It can be shown that if a solution exists, then () is the expected value of () at the (random) first exit point from for a canonical Brownian motion starting at .
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
The boundary conditions on the function (,) and its normal derivatives / are imposed at the edges of the surface patch. With the above formulation it is notable that the elliptic partial differential operator in the above PDE represents a smoothing process in which the value of the function at any point on the surface is, in some sense, a ...