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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).
In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem.It involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem.
If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that G(x,x′) vanishes when either x or x′ is on the bounding surface. Thus only one of the two terms in the surface integral remains. If the problem is to solve a Neumann boundary value problem, it might seem logical to choose Green's ...
In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani 's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion .
The boundary value problem solver's performance suffers from this. Even stable and well-conditioned ODEs may make for unstable and ill-conditioned BVPs. A slight alteration of the initial value guess y 0 may generate an extremely large step in the ODEs solution y(t b; t a, y 0) and thus in the values of the function F whose root is sought. Non ...
In dynamical problems with periodically varying forces, the equation of motion sometimes takes the form of Mathieu's equation. In such cases, knowledge of the general properties of Mathieu's equation— particularly with regard to stability of the solutions—can be essential for understanding qualitative features of the physical dynamics. [ 41 ]
The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0, L 1 or y = 0, L 2 and define the simplest possible Sturm–Liouville eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence, (,,) = where m and n are non-zero integers, A mn are arbitrary constants ...
Inspired by the recent successful applications of the HAM in different fields, a Mathematica package based on the HAM, called BVPh, has been made available online for solving nonlinear boundary-value problems .