On the Reduction of a System of Partial Differential Equations to Systems of Ordinary Differential Equations

   The system of Maxwell’s equations is considered in the Darwin approximation. The study of the system is based on the reduction of systems of partial differential equations (linear and nonlinear) to systems of ordinary differential equations. The variable ψ, where ψ(x, y, z, t) = const is the level surface of some functions, is chosen as an independent variable in systems of ordinary differential equations. The reduction is based on constructing an extended system of equations of characteristics (basic system) for a partial differential equation of the first order (basic equation), which is satisfied by the level surface of the selected functions. All the necessary relations are added to the basic system as the first integrals to obtain a system of ordinary differential equations for the system of partial differential equations under consideration. To search for level surfaces for solutions of the system of equations under consideration, both the approaches previously described in a number of our articles and the newly proposed variants of our method are used. Three systems of ordinary differential equations with different independent variables (different functions ψ(x, y, z, t)) are presented. It is shown that obtaining a level surface in each of the considered approaches has a functional arbitrariness. Some exact solutions of the considered system of partial differential equations are obtained. As an example, for one of the systems of ordinary differential equations, the set of solutions of which depends on the choice of an arbitrary function g(ψ) in the basic equation t = g(ψ), a solution is written for the case where g(ψ) = ψ. The solution of the problem of determining a vortex-free electromagnetic field by a given charge
distribution is given.

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