Connection between numerical integration and solving ODE numerically.

[riemann86]

Hey

I have taken a programming course. And I have learned about Simpson, Trapezoidal and the midpoint rule etc, I have programmed these.
I have also implemented forward Euler, backward euler, Runge Kutta etc for solving ODE.

I am wondering if there is any way to unify these two things, are there any texts you know that treat these two things as allmost the same? They were thought seperately but when you solve an ODE analytically, you integrate. So the Euler method is kind of an integration, but these methods are not derived this way, they are derived with you wanting to iterate forward, and it is not an emphasis on integration.

Are there any texts or anything that unites these two things? It seems like the only difference is that in one case you have to iterate and in the other you don't really iterate, but in both you integrate?

 

[Office_Shredder]

Imagine the following differential equation: dy/dx = f(x)
Doing eulers method is the same as integrating f(x) using left hand sums. You should check this as an exercise

In general a way of solving odes can be reduced to integrating a function by this specialization, but it's not worth mentioning this because solving odes typically requires a much more general way of iterating forward than integration does, because when integrating you already know the derivative everywhere

 

[riemann86]

Thank you very much, I think I understand it much clearer now.
Would it be fair to say then that a problem of solving an ODE will be reduced to numerical integration if one actually knew the exacat derivative at each point?, but it is not the same since we must approximate the derivative?

I checked with your example by writing the sums and it works, thanks! :)
But is it so that there for example exists a numerical way of solving ODE that will boil down to simpson if we actually had the exact derivative at each point?
And can we go the opposite way, is there a way to simplify the Runge Kutta method for example, so that if the function is known, we can use Runge Kutta to integrate it?

 

[HallsofIvy]

when you solve an ODE analytically, you integrate.

This is true only for a limited variety of first order differential equations. For example, to solve the comparatively simple y''+ y= 0, y(0)= 0, y(1)= 0, you do not integrate.

 

[blue_raver22]

Yes, there is a connection between numerical integration and solving ODEs numerically. In fact, numerical integration methods are often used to solve ODEs numerically. This is because solving an ODE involves finding a function that satisfies the ODE, whereas numerical integration involves finding an approximation of the area under a curve. Both of these tasks involve finding a function, and therefore, there are similarities between the two.

One way to think about this connection is to consider that solving an ODE is essentially finding the anti-derivative of the ODE, which is the same as finding the area under the curve represented by the ODE. This is where numerical integration methods come in - they provide a way to approximate this area under the curve, which in turn helps us find a solution to the ODE.

There are many texts and resources that discuss this connection and approach numerical integration and solving ODEs in a unified manner. Some examples include "Numerical Methods for Engineers" by Steven Chapra and Raymond Canale, "Numerical Recipes" by William H. Press et al., and "Numerical Analysis" by Timothy Sauer. These texts explain how numerical integration methods can be used to solve ODEs and provide examples and exercises to help reinforce this connection.

Overall, understanding the connection between numerical integration and solving ODEs numerically can help deepen your understanding of both topics and make it easier to apply these methods in your work as a scientist.