Term
Fundamental Solution Matrix 

Definition
For a linear BVP, let y_{i}(t) be the solution to the homogeneous ODE y' = A(t)y with initial condition y(a) = e_{i}. Let Y(t) denote the matrix whose ith column is y_{i}(t). Y is the fundamental solution matrix for the ODE, and its columns are called solution modes. 


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Definition
y'=A(t)y + b(t), a < t < b where A(t) and b(t) are continuous, with boundary conditions B_{a}Y(a) + B_{b}Y(b) = c has a unique solution iff the matrix Q = B_{a}Y(a) + B_{b}Y(b) is nonsingular. 


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Definition


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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 wellconditioning, the growing and decaying modes must be controlled appropriately by the boundary conditions imposed. 


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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. 


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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) = ∑x_{i}φ_{i}(t) where φ_{i} are basis functions defined on [a,b] and x is an nvector of parameters to be determined. 


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Definition

