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Is there some method for solving differential equations that can be applied to Maxwell equations to always get a solution for the electromagnetic field, even if numerical, regardless of the specifics of the problem.

Let's say you want to design a series of steps that you can handle to a student and he will be able to obtain E and B for any problem. The instructions don't have to be simple or understandable to someone without proper background but, is it possible?

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    $\begingroup$ This is extremely vague. There are many different ways of posing problems. You could have a given set of sources, you could have boundary conditions in which the fields are given on a certain surface, you could have conducting objects, and so on. It also isn't obvious what you mean by "solving" Maxwell's equations. Do you want a closed-form solution in terms of a certain set of functions? Are you satisfied with a numerical solution? $\endgroup$ – user4552 Oct 4 '19 at 13:34
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    $\begingroup$ FEMM anyone? $\endgroup$ – ja72 Oct 4 '19 at 21:49
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You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.

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  • $\begingroup$ Wow this is super cool! $\endgroup$ – user541686 Oct 4 '19 at 8:02
  • $\begingroup$ This should be the accepted answer... $\endgroup$ – Rexcirus Oct 4 '19 at 13:45
  • $\begingroup$ This is very useful. Thank you. $\endgroup$ – Chegon Oct 4 '19 at 21:17
  • $\begingroup$ In the same link, Jeffimenko equations are derived from the simpler retarded potential integrals. These can also be used in a numerical solution if the sources and time are known. $\endgroup$ – Riad Dec 10 '19 at 12:29
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Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).

The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.

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  • $\begingroup$ This doesn't look like a "general way" — this only explains that the solution exists given the initial conditions. If you have a complicated initial condition given for infinite space, you won't immediately be able to take any available numerical PDE solver and just apply it. $\endgroup$ – Ruslan Oct 4 '19 at 5:49
  • $\begingroup$ Maxwell equations are linear, so you can super impose simpler situations to get to the final answer. In any case ill posed initial conditions can't give a solution. $\endgroup$ – Riad Dec 10 '19 at 11:57
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Exact solution of Maxwells equations does not exist (for whole space and time). Maxwells equations describe behaviour of the fields localy (in a neighbourhood of a point) because to calculate any field in a point, all you need is the information about it's neighbourhood. If you place random objects that modify the $E$ field or $B$ field, you have to take those into account, but there is no way to embed them to those equations other that to say that those fields have to satisfy some additional conditions (i.e boundary conditions) on those objects. Analytical solving often assumes special cases (e.g free space, $\rho=0, J=0$) to simplify the equations. Numerically, you can allways find better and better approximation of the distribution of fields in space by using smaller steps in space and time during discretization. Finate difference method and it's variants are often used for solving differential equations in general.

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    $\begingroup$ Given sufficiently smooth initial conditions the exact solution does exist. It may be impossible to write in closed form, but that doesn't result in inexistence of the solution. $\endgroup$ – Ruslan Oct 4 '19 at 5:21
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    $\begingroup$ This doesn't make much sense. Exact solutions of Maxwell's equations certainly exist, e.g., a plane wave. $\endgroup$ – user4552 Oct 4 '19 at 13:31
  • $\begingroup$ Maxwell equations are hyperbolic(initial value) and for this reason they don'e have a general solution for the whole space and all the time. The introduction of a new charge particle is not a problem if you bring it from infinity. This is why Einstein changed the wave equation using imaginary time to get an elliptic equation that could have a solution for all space-time. $\endgroup$ – Riad Dec 10 '19 at 12:02

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