%0 Journal Article
%T Lie Symmetry Analysis of 1D Proper-Time Maxwell¡¯s Equations: Exact Solutions and Conservation Laws<br />
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%A Joshua Owolabi Adeleke
%J Open Access Library Journal
%V 12
%N 10
%P 1-12
%@ 2333-9721
%D 2025
%I Open Access Library
%R 10.4236/oalib.1114307
%X We present a complete Lie symmetry analysis of the one-dimensional proper-time formulation of Maxwell¡¯s equations with velocity-dependent propagation speed <em>b=</em><span style="font:15px Tahoma;"><em>¡Ì</em></span><span style="font:13px Tahoma;border-top:1px solid black;"><em>c</em><sup><em>2</em></sup>&#43;u<sup><em>2</em></sup></span>. Through systematic application of symmetry methods, we: 1) classify all Lie point symmetries of the system, 2) derive exact invariant solutions via symmetry reduction, and 3) construct conserved quantities using Noether¡¯s theorem. The solutions exhibit characteristic propagation at speed <em>b</em> , maintaining relativistic causality through the constraint <em>u</em><sup><em>¦Ì</em></sup><em>u</em><sub><em>¦Ì</em></sub><em>=-c</em><sup><em>2</em></sup>. Numerical verification confirms solution stability under appropriate discretization. This work establishes a rigorous mathematical foundation for proper-time electrodynamic systems, with applications to particle acceleration and high-energy astrophysical phenomena. The novelty lies in the application of these methods to the proper-time formulation, revealing new solution structures tied to the relativistic propagation speed <em>b </em>.
%K Lie Symmetry Analysis
%K Proper-Time Formulation
%K Maxwell¡¯s Equations
%K Conservation Laws
%K Exact Solutions
%U http://www.oalib.com/paper/6875250