In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional
version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make
the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational
method based on higher-order spline interpolations is used to approximate the solution relatively to
the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are
studied and proved, and the study is also provided by an error estimate. Also, we use the critical
Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and
for all numerical computations, we consider several quadrature methods. We give some experiments
to illustrate the success of such an approach. Finally, we apply the higher-order spline interpolants
to solve the first kind of Fredholm integral equation and Pocklington’s integro-differential equation to
treat the signal reconstruction inside the wire antennas.