Abstract
In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms.
Highlights
Since this paper is focused on solving non-stationary problems, it is appropriate to modify Maxwell’s equations into a more suitable form for optimization, qualitative analysis, and numerical methods
The use of this procedure for qualitative analysis, numerical methods, and optimization are presented in the individual Sections for both the vector variant and the scalar variant of Maxwell’s equations
The work was focused on the analysis of non-stationary Maxwell equations
Summary
Since this paper is focused on solving non-stationary problems, it is appropriate to modify Maxwell’s equations into a more suitable form for optimization, qualitative analysis, and numerical methods This can be achieved by using Gauss’s divergence theorem and the symmetry of the Kronecker delta operator δij = δji. It follows from the above-mentioned equations that the article focuses on the appropriate modification of Maxwell’s equations so that the influence of non-stationary terms in the field V is expressed by their values on the boundary S of the closed region. The solution is based on the use of symmetry conditions and Gauss’s divergence theorem The use of this procedure for qualitative analysis, numerical methods, and optimization are presented in the individual Sections for both the vector variant and the scalar variant of Maxwell’s equations. This method does not allow the use of the finite volume method, while the new variant will allow it; see Section 2
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