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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 ...
Kansa method has recently been extended to various ordinary and PDEs including the bi-phasic and triphasic mixture models of tissue engineering problems, [14] [15] 1D nonlinear Burger's equation [16] with shock wave, shallow water equations [17] for tide and current simulation, heat transfer problems, [18] free boundary problems, [19] and ...
The MFS has proved particularly effective for certain classes of problems such as inverse, [10] unbounded domain, and free-boundary problems. [11] Some techniques have been developed to cure the fictitious boundary problem in the MFS, such as the boundary knot method, singular boundary method, and regularized meshless method.
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 ...
Using the SAT technique, the boundary conditions of the PDE are imposed weakly, where the boundary values are "pulled" towards the desired conditions rather than exactly fulfilled. If the tuning parameters (inherent to the SAT technique) are chosen properly, the resulting system of ODE's will exhibit similar energy behavior as the continuous ...
The solution to a boundary problem with generalized boundary conditions is solvable whenever the comparison principle holds. [ 4 ] The stability of solutions in L ∞ {\displaystyle L^{\infty }} holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution).
The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases ...
A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge–Ampere equation.