Search results
Results from the WOW.Com Content Network
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).
Thus, one of the two sets of boundary functions {f 1, f 2} or {g 1, g 2} is redundant, and this implies that a partial differential equation with boundary conditions on a rectangle cannot have arbitrary boundary conditions on the borders, since the conditions at x = x 1, x = x 2 must be consistent with those imposed at y = y 1 and y = y 2.
Thus, solutions of the boundary value problem correspond to solutions of the following system of N equations: (;,) = (;,) = (;,) =. The central N−2 equations are the matching conditions, and the first and last equations are the conditions y(t a) = y a and y(t b) = y b from the boundary value problem. The multiple shooting method solves the ...
Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves ...
Robin boundary conditions are commonly used in solving Sturm–Liouville problems which appear in many contexts in science and engineering. In addition, the Robin boundary condition is a general form of the insulating boundary condition for convection–diffusion equations. Here, the convective and diffusive fluxes at the boundary sum to zero:
The differential equation is said to be in Sturm–Liouville form or self-adjoint form.All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector).
Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary. For example, given a solution u to a partial differential equation on a domain Ω with boundary ∂Ω, it is said to satisfy a mixed boundary condition if ...
Since the problems in magnetostatics involve solving Laplace's equation or Poisson's equation for the magnetic scalar potential, the boundary condition is a Neumann condition. In spatial ecology , a Neumann boundary condition on a reaction–diffusion system , such as Fisher's equation , can be interpreted as a reflecting boundary, such that ...