Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \leq
t \leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently.
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##problem 2
Implement an explicit Euler method to solve ODE's. The explicit euler method has the parameters $c_i = 0$, $a_{ij} = 0$ and $b_1 = 0$ which then yields the update formula
$$y_{n + 1} = y_n + h f ( t_n , y_ n )$$
Choose a value h for the size of every step and set $t_n = t_0 + n h$ .
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## problem 3
*stiff* ODE's are somewhat *ill-conditioned* in the sense, that many algorithms for solving this type of ODE are then unstable. Consider the initial value problem:
$$y^\prime ( t ) = - 15 y ( t ) , t \ge 0 , y ( 0 ) = 1$$
and solve it with the `odeSolve` and the explicit euler. Look especially at $h= 1/4$ and $h = 1/8$ for the euler method and look what is happening.
Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \leq
t \leq 3$. Use at least two different solvers and compare their results.
t \leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently.
*stiff* ODE's: Consider the initial value problem:
*stiff* ODE's are somewhat *ill-conditioned* in the sense, that many algorithms for solving this type of ODE are then unstable. Consider the initial value problem:
$$y^\prime ( t ) = - 15 y ( t ) , t \ge 0 , y ( 0 ) = 1$$
and solve it with the odesolve
and solve it with the `odeSolve` and the explicit euler. Look especially at $h= 1/4$ and $h = 1/8$ for the euler method and look what is happening.