# Shared Flashcard Set

## Details

Boundary Value Problems for ODEs
Shooting Method; Finite Difference Method; Collocation Method; Galerkin Method
9
Computer Science
03/05/2014

Term
 Fundamental Solution Matrix
Definition
 For a linear BVP, let yi(t) be the solution to the homogeneous ODE y' = A(t)y with initial condition y(a) = ei.  Let Y(t) denote the matrix whose ith column is yi(t).  Y is the fundamental solution matrix for the ODE, and its columns are called solution modes.
Term
 Unique Solution BVP
Definition
 y'=A(t)y + b(t), a < t < b where A(t) and b(t) are continuous, with boundary conditions BaY(a) + BbY(b) = c has a unique solution iff the matrix Q = BaY(a) + BbY(b) is nonsingular.
Term
 Green's Function
Definition
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Term
 Dichotomy
Definition
 Conditioning/Stability depends on the interplay between the growth of solution modes and the boundary conditions.  The solution is determined everywhere at once (rather than looking for something growing with time).  To be well-conditioning, the growing and decaying modes must be controlled appropriately by the boundary conditions imposed.
Term
 Shooting Method
Definition
 Replace BVP with sequence of IVPs.  Guess a value for initial slope, solve the resulting IVP and then check to see if the computed solution value at t=b matches desired boundary value.
Term
 Finite Difference Method
Definition
 Convert BVP directly into system of algebraic equations rather than a sequence of IVPs.  Introduce equally spaced mesh points.  Replace the derivatives appearing in the ODE by finite difference approximations.  System of equations can determine the approximate solution at all points simultaneously.
Term
 Finite Difference Method -- Nonlinear System
Definition
 If the system obtained by the finite difference method is nonlinear, an iterative method (such as Newton's method) is required to solve it in which case a reasonable starting guess is to choose values on a straight line between (a, alpha), (b, beta).
Term
 Collocation Method u'' = f(t, u, u'), a < t < b u(a) = α, u(b) = β
Definition
 We seek an approximate solution of the form u(t) = v(t,x) = ∑xiφi(t) where φi are basis functions defined on [a,b] and x is an n-vector of parameters to be determined.
Term
 Galerkin Method
Definition
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