Numerically Solving ODE with Lagrange Multipliers

[davidkais]

Hi,

I'm trying to implement some equations from a paper. It comes down to a system of 2 coupled ODEs. In one of the ODEs, there are 3 Lagrange multipliers. The paper says that the three multipliers can be determined by three integral constraints (integrals of some functions of the solutions of the ODEs are equal to some value). I don't get how this can be numerically solved with any degree of efficiency. It appears to solve the ODEs you need to know the value of the Lagrange Multipliers, but to evaluate the Lagrange multipliers you need to know the solution of the ODE. My supervisor isn't too forthcoming with help - it seems he's very busy. The only way I can think of this solving this is to a try thousands of random multipliers and retain the multipliers which return the smallest error against the three defined constraints. Is there some common magical numerical technique for solving equations of this type? I presume there is as the paper from which I got the equations describes this part in little detail.

I can provide the functions and integrals in question, if necessary. I apologise if any parts were unclear. As you probably know, it's sometimes hard to describe maths with words.

Thanks,
David

 

[jvicens]

Hi David,

I can understand your confusion with trying to solve a system of coupled ODEs with three Lagrange multipliers. This can be a challenging task, but there are some techniques that can make it more efficient.

One approach is to use numerical methods such as Runge-Kutta or Euler's method to solve the ODEs and then use a root-finding algorithm to determine the values of the Lagrange multipliers that satisfy the integral constraints. This involves repeatedly solving the ODEs with different values of the multipliers until the constraints are met.

Another approach is to use optimization techniques such as gradient descent or Newton's method to simultaneously solve the ODEs and determine the values of the multipliers that minimize the error between the solutions and the constraints. This can be a more efficient method, but it requires some understanding of optimization algorithms and may require some trial and error to find the best approach.

In either case, it would be helpful to have the specific functions and integrals in question so that the best approach can be determined. It's also a good idea to consult with your supervisor or a colleague who may have experience with solving similar equations.

I hope this helps and good luck with your research!