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3

Your profile lists your age as 17, so I assume you're still at school. At this stage in your physics education you'll only have been exposed to differential equations that have relatively straightforward solutions. I assume the education system does this to avoid putting you off. If you continue studying physics you'll quickly learn that the vast majority of ...


3

There is a mathematical point that can be made, and in my opinion is related to a deeper understanding of what it means to solve a (partial) differential equation. I will try to keep things simple, and consider only linear models. Suppose that you have a space $X$ with some properties, for example it has a topology. We suppose that the state of our system ...


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Delay equations are not necessarily approximations. They often appear if a projection is made, see http://www.physicsoverflow.org/17968/how-to-handle-nonmarkovian-dynamics-in-open-quantum-system


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Don't do it that way. You have what is called a "Change Point". Run it up until the time when the change should occur. Then stop the solver. Perform the instantaneous state change. Then restart the solver. So much silliness happens when people try to run ODE solvers over discontinuities.


0

RK 4th order is a good numerical approach - but it is only accurate up to fourth order terms in the Taylor expansion of your series. As long as the fifth (and higher) order derivatives of the function are small, you are fine. But when you introduce a step function, or even a piecewise linear approximation, that assumption is violated. I would recommend, as ...


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I don't think that you need some complicated apparatus. You can simply solve this problem by writing $$ \dot{c}_k(t) = M_{kj} c_j(t). $$ Now because $M$ is symmetric and real you can find a transformation $\tilde{c}_k$ of the $c_k$ that diagonalizes $M$ and leads to trivial differential equations for the transformed functions.


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How accurate are differential equations really, and to what accuracy can we predict future circumstances and events from them? Why does it matter that they are "differential equations"? Differential equations are just one type of model. The question is how accurate are theoretical models. The answer is, the ones you learn about in high school / college ...



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