> Hi,
>>On 01/20/2014 01:55 AM, Arijit Hazra wrote:
>> Hello All,
>>
>>I am trying to solve an ode with time-dependent parameters by using
>> boost. I have just started using boost and I think boost library's
>>odeint library is the best option for my problem.
>>
>> Would anyone please explain me how I can solve a Initial Value Problem
>>--a system of ODE --with a time-dependent parameter. e.g.
>>
>> | dy1/dt = a1(t) * y2 -b1(t)* y3 -c1(t) y1;
>> dy2/dt = a2(t) * y1 -b2(t)* y1 -c2(t) y3;
>> dy3/dt = a3(t) * y2 -b3(t)* y3 -c3(t) y1;
>> |
>>
>> y(0)=c; My idea so far after going through the manuals and help files.
>> is that for fixed time-step if I form a vector of a,b,c and assume a,b,c
>> to be constant over that time-period I can solve using boost library.
>How do a,b,c look like? Are it functions you can evaluate, like sin(),
>cos(), .... or combinations of it? If this the case you can just write
>down the equations like
my parameters are basically a1(t,params) ,a2(t,params), a3(t,params) are piecewise continuous functions dependent on t and some other fixed parameters e.g. it is sinc function in some interval,>>On 01/20/2014 01:55 AM, Arijit Hazra wrote:
>> Hello All,
>>
>>I am trying to solve an ode with time-dependent parameters by using
>> boost. I have just started using boost and I think boost library's
>>odeint library is the best option for my problem.
>>
>> Would anyone please explain me how I can solve a Initial Value Problem
>>--a system of ODE --with a time-dependent parameter. e.g.
>>
>> | dy1/dt = a1(t) * y2 -b1(t)* y3 -c1(t) y1;
>> dy2/dt = a2(t) * y1 -b2(t)* y1 -c2(t) y3;
>> dy3/dt = a3(t) * y2 -b3(t)* y3 -c3(t) y1;
>> |
>>
>> y(0)=c; My idea so far after going through the manuals and help files.
>> is that for fixed time-step if I form a vector of a,b,c and assume a,b,c
>> to be constant over that time-period I can solve using boost library.
>cos(), .... or combinations of it? If this the case you can just write
>down the equations like
>double a1( double t ) { // return a1(t); }
>double a2( double t ) { // return a1(t); }
// ...
>void ode( state_type const &y , state_type &dydt , double t )
>{
> dydt[0] = a1(t) * y[1] - b1(t) * y[2] - c1(t) * y[0];
> // ...
>}
e.g.
double a1(double t, double c1, double c2)
{
//
{
//
return a1;
}
similarly for a2 and a3. i am confused how I can pass the a1,b1,c1 and so on in the ODE function and solve it.
Suggestions would be helpful.
>The steppers in odeint do not assume that a,b,c are constant over the
>time step. If you use for example a multi-stage method like Runge-Kutta
>it will evaluate a,b,c at different times during one time step. It also
>works for controlled stepper with step size control. Ff course, the
>stepper do not know what happens between the evaluations - this
>inaccuracy is the price you have to pay when solving ODEs numerically,
>but it should not be too large and can be controlled when using step
>size adaption.
>time step. If you use for example a multi-stage method like Runge-Kutta
>it will evaluate a,b,c at different times during one time step. It also
>works for controlled stepper with step size control. Ff course, the
>stepper do not know what happens between the evaluations - this
>inaccuracy is the price you have to pay when solving ODEs numerically,
>but it should not be too large and can be controlled when using step
>size adaption.
> But this is mathematically less accurate and computationally expensive
> as far my understanding .For solving an ODE numerically
> evaluation for explicit methods and Jacobian evaluation for Implicit
> methods at specific points in /t/ and /y/.>
> e.g a step of explicit Euler is /yn/+1=/yn/+(/tn/+1−/tn/)⋅/f/(
/yn/,/tn/).
>
> For time-dependent parameter, treating it as part of /f/, the function
>can safely use this parameter to solve a non-autonomous ODE. Of course,
>this is exactly equivalent to solve the enhanced ODE with an additional
>dimension for the time, in your example x1 = y1 , x2= y2 , x3 = y3 , x4 = t;
>therefore
>dx1/dt = a1(x4) * x2 - b1(x4) * x3 - c1(x4) * x1;
// ...
Hth,
>Different algorithms are derived to solve different ways, some assume a, b and c are constant over the whole step, and >some require that you update a, b and c during the substeps of the calculation (e.g. multistage). Typically, algorithms >assume you are going to do the "right thing" and that whenever they ask you to evaluate dydt, you're evaluating that >vector with the most recent information. You don't *have* to do that in a multistage algorithm, but as the SciComp answer >suggested, if you do assume your parameters are constant, you're effectively wasting the advantages of a multistage >method.
>
> For time-dependent parameter, treating it as part of /f/, the function
> evaluation (respectively, for implicit methods, also treat it as part of
> the Jacobian evaluation) is the standard procedure. A similar strategy
> applies to more complicated methods for solving ODEs (multistage methods
> such as Runge-Kutta, implicit methods for stiff systems, etc.).
>Nearly all steppers pass the time explicitly to the ODE, see above. You> the Jacobian evaluation) is the standard procedure. A similar strategy
> applies to more complicated methods for solving ODEs (multistage methods
> such as Runge-Kutta, implicit methods for stiff systems, etc.).
>can safely use this parameter to solve a non-autonomous ODE. Of course,
>this is exactly equivalent to solve the enhanced ODE with an additional
>dimension for the time, in your example x1 = y1 , x2= y2 , x3 = y3 , x4 = t;
>therefore
>dx1/dt = a1(x4) * x2 - b1(x4) * x3 - c1(x4) * x1;
// ...
Hth,
>Different algorithms are derived to solve different ways, some assume a, b and c are constant over the whole step, and >some require that you update a, b and c during the substeps of the calculation (e.g. multistage). Typically, algorithms >assume you are going to do the "right thing" and that whenever they ask you to evaluate dydt, you're evaluating that >vector with the most recent information. You don't *have* to do that in a multistage algorithm, but as the SciComp answer >suggested, if you do assume your parameters are constant, you're effectively wasting the advantages of a multistage >method.
Now, I understood the mathematical parts I need though it is far from rigorous knowledge.Thanks for your cooperation.
>Is this homework? Do you have to do this in C++ (which is what Boost is written in), or can you choose any language? >Do you actually have to program it?
No this is not a homework problem. I am trying to build a code with solving the system of ODE I mentioned as the first building block.I think odeint library of boost is best for my purpose.
Thank you for your co-operation.
Arijit