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
|
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
|
Definition
|
|
Term
|
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
|
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
|
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
|
Definition
|
|
