Abstract
In this chapter we study initial value problems for systems of ordinary differential equations. We also consider some boundary value problems. Such differential systems occur often in science and engineering. Existence and uniqueness of a solution can be proved under general conditions, but it is in general impossible to write down the exact solution. Numerical methods (ODE solvers) provide tools to obtain discrete approximations. We give examples of differential systems and some numerical methods for their approximation.
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Notes
- 1.
The solution is the sum of the general solution of the homogenous equation \(y'(t) = -150y(t)\) and a particular solution that is a linear polynomial.
- 2.
It is a geometric sequence.
- 3.
Show an equality like (12.8).
- 4.
See Chap. 3.
- 5.
Construct the solution row by row.
- 6.
Use (12.6) and the Lipschitz condition of f.
- 7.
Use \(f(t,\boldsymbol{y}(t)) = \boldsymbol{y}'(t)\) and introduce \(\boldsymbol{y}'(t + h) - f\big(t + h,\boldsymbol{y}(t + h)\big)\) in \(\phi (t,\boldsymbol{y}(t),h)\).
- 8.
Use (12.6) and the Lipschitz condition of \(f\).
- 9.
Compute E′(t)
- 10.
Integrate y′(t)∕y(t).
- 11.
\(p \in \mathbb{P}_{2}\).
- 12.
The exact solution is y(t) = sin(t).
- 13.
Use grid on.
- 14.
Since n is modified, σ 1 has to be recomputed.
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Lyche, T., Merrien, JL. (2014). Ordinary Differential Equations, One Step Methods. In: Exercises in Computational Mathematics with MATLAB. Problem Books in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43511-3_12
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