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Definition
Highestorder derivative appearing in the ODE 


Term

Definition
A higherorder ODE can always be transformed into equivalent firstorder system. For explicit kth order, define k new unknowns u_{1}(t) = y(t), ..., u_{k}(t) = y^{(k1)}(t).
Can write as system u' = g(t,u) where u_{1}^{'}=u_{2}, ..., u_{k}' = f(t, u_{1}, ..., u_{k}) 


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If f does not depend explicitly on t
Can be written in form y' = f(y) 


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Linear Homogeneous Systems with Constant Coefficients 

Definition
Linear  f has form f(t,y) = A(t)y + b(t) where A(t) and b(t) are matrixvalued and vectorvalued functions of t
Homogeneous  b(t) = 0
Constant Coefficients  A does not depend on t



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Definition
Solution to u' = f(y) is stable if for every ε > 0, there is a δ > 0 such that if y(t) satisifies the ODE and y(t)  u(t) <= δ then y(t)  u(t) <= ε for all t. 


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Jacobian matrix for nonlinear system 

Definition
{J_{f}(t,y)}_{ij} = ∂f_{i}(t,y)/∂y_{j} 


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Definition
y_{k+1}=y_{k}+h_{k}f(t_{k},h_{k}) 


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Definition
y_{k+1}=y_{k}+h_{k}f(t_{k+1},y_{k+1}) 


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Definition
y_{k+1}=y_{k}+h_{k}(f(t_{k},y_{k}) + f(t_{k+1}, y_{k+1}))/2 


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Definition
Due to the method used, and would remain even if all arithmetic were performed exactly 


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Definition
The error made in one step of the numberical method 


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Definition
Cumulative overall error. The error at step k is the difference between the approximate solution, y_{k}, and the true solution y(x_{k}). 


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Definition
Accuracy of a numerical method is of order p if l_{k}=O(h_{k}^{p+1}) 


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Stability of Numerical Method 

Definition
Small perturbations do not cause the resulting numerical solution to diverge away without bound 


Term
Euler's Method for Scalar ODE y'=λy 

Definition
y_{k+1}=y_{k}+h λy_{k}
y_{k+1}=(1+ hλ)^{k}y_{0}
So (1+ hλ)^{k} is called the growth factor, and the magnitude of 1+ hλ must be less than one with Re(λ)<0, otherwise the method is unstable 


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Definition


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Definition
The global error is not simply the sum of the local errors. If the solutions of the ODE are diverging, the local errors at each step are magnified over time, so that global error is greater than sum of local error. If solutions of ODE are converging then global error may be less than that sum fo the local errors. 


Term

Definition
y' = λy, y(0) = y_{0}
has exact solution y(t) = y_{0}e^{λt}
Euler's Method: y_{k+1} = y_{k} + λhy_{k} = (1 + λh)y_{0}
So, (1 + λh) is an amplification factor (must be less than 1 in magnitude) 


Term

Definition
Use information at more than one previous point to estimate the solution at the next point. Need to use some other method to obtain the initial point. 


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Adams Explicit Method (AdamsBashforth predictor) 

Definition


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Adams Implicit Method (AdamsMoulton Corrector) 

Definition


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Definition
A good initial guess is conveniently supplied by an explicit method so the explicit and implicit methods are used together. A fixed number of corrector steps (often only one) can be used to reduce the expense of implicit methods. 


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Derived by interpolating derivative values y' = f at m previous points and then integrating the resulting interpolating polynomial to obtain
[image] 


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Backwards Differentiation Formula Methods 

Definition
Derived by interpolating the solution values y at m previous points, differentiating the resulting interpolating polynomial, and setting the derivative equal to f(t_{k+1}, y_{k+1}) at t_{k+1} to obtain y_{k+1}. 


Term

Definition
Retaining more terms in the Taylor series, we can generate higherorder singlestep methods than Euler's method
e.g.: y_{k+1}=y_{k}+h_{k}y_{k}' + h_{k}^{2}/2 y_{k}'' 


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Definition
Singlestep methods, similar to Taylor series methods, but replace higher derivatives by finite difference approximations based on values of f at points between t_{k} and t_{k+1} 


Term

Definition
Based on use of a singlestep method to integrate the ODE over a given interval using several different step sizes and yilding results denoted by Y(h_{i}). Gives discrete approximate to Y(h) where Y(0) = y(t_{k+1}). Fit interpolating polynomial to these data, and approximate Y(0). 

