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Introduction to the Numerical Solution of IVP for ODE
Introduction to the Numerical Solution of IVP for ODE
Consider the IVP: DE x = f (t, x), IC x(a) = xa . For simplicity, we will assume here that
x(t) Rn (so F = R), and that f (t, x) is continuous i
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Ordinary Dierential Equations
Linear ODE
Let I R be an interval (open or closed, nite or innite at either end). Suppose
A : I Fnn and b : I Fn are continuous. The DE
( )
x = A(t)x + b(t)
is called a rst-order linear [system of] ODE[s] on I . Since f (t
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Continuity and Dierentiability of Solutions
Continuity and Dierentiability of Solutions
We now study the dependence of solutions of initial value problems on the initial values and
on parameters in the dierential equation. We begin with a fundamental e
Ordinary Dierential Equations
Existence and Uniqueness Theory
Let F be R or C. Throughout this discussion, | | will denote the Euclidean norm (i.e 2 norm) on Fn (so is free to be used for norms on function spaces). An ordinary dierential
equation (ODE) is