"Solve the system with initial value $(1,0,0)$ at $0$ with the time range$0 \\leq\n",

"t \\leq 3$. Use at least two different solvers and compare their results. Try to understand heuristicly why different solvers behave sometimes very differently."

],

"cell_type": "markdown"

},

{

"metadata": {},

"source": [

"##problem 2\n",

"\n",

"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\n",

"\n",

"$$y_{n + 1} = y_n + h f ( t_n , y_ n )$$\n",

"\n",

"Choose a value h for the size of every step and set $t_n = t_0 + n h$ . \n"

],

"cell_type": "markdown"

},

{

"metadata": {},

"source": [

"## problem 3\n",

"\n",

"*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: \n",

"\n",

"$$y^\\prime ( t ) = - 15 y ( t ) , t \\ge 0 , y ( 0 ) = 1$$\n",

"\n",

"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.\n",

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.

%% Cell type:markdown id: tags:

##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 $t_n = t_0 + n h$ .

%% Cell type:markdown id: tags:

## 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\n",

"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."

]

},

{

...

...

@@ -131,11 +131,11 @@

"source": [

"## problem 3\n",

"\n",

"*stiff* ODE's: Consider the initial value problem: \n",

"*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: \n",

"\n",

"$$y^\\prime ( t ) = - 15 y ( t ) , t \\ge 0 , y ( 0 ) = 1$$\n",

"\n",

"and solve it with the odesolve\n",

"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.\n",

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.