Sumudu Characterization of the Maxwell Eigenvalue Problem

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In this project we investigate the distinction ensued by adding an eigenvalue Magnetically related forcing term, on the dynamics of a planar, transverse electromagnetic (TEMP) wave propagating in the direction z in lossy media with constant permittivity ε, permeabilityµ, and conductivity σ > 0. In the presence of the magnetic eigenvalue forcing term, the TEMP is then best described by the Modified Maxwell's equations, or Maxwell Eigenvalue Problem, (1) When the source term, m (z, t) = 0, various forms of solutions for the electric and magnetic fields have been determined by various authors, using different techniques. Recently, many aspects of Maxwell TEMP Connoted problem were resolved by using the Sumudu transform technique (See for instance, Belgacem, Husein-Belgacem, Rathinavel-Belgacem). The Sumudu transform technique is not only robust due to its various desirable attributes such as scale and units preserving properties, elaborated upon in the body of the project, but also it turns out to be user friendly so to speak. In this instance we use the Sumudu to variationally characterize the principal eigenvalue of Maxwell's eigenproblem as was done for the Indefinite Ecological problem by Belgacem and Belgacem-Cosner. The function, being the Sumudu of is obtained by integartion against the kernel, e-1, as follows, (2) As an operator, the domain of the Sumudu extends over the set of functions, (3) For instance, we have, The Sumudu Operator exhibits very direct convolution, properties, namely, when M(u) and N(u), are the respective Sumudi for the functions f(t) and g(t) , with convolution (f*g)(t).then, the Sumudu of the convolution of the functions, f(t), and g(t), is given by, S[(f*g)(t)] . In particular, the Sumudu of the anti-derivative, (f * 1), of the function, f(t) , is given by; uM(u). The Sumudu of the n'th derivative, some of the main results obtained via the Sumudu method are, Theorem 1 [Husein-Belgacem]: For , the transient electric field, E(z,t), in the TEMP problem as described in equation (1), is given by, Theorem 2 [Rathinavel-Belgacem]: For , the transient electric field, H(z,t), in the TEMP problem as described in equation (1), is given by, In this project we extend theorems 1 & 2 to cases where the source term is acting positively (a positive source) and uniformly, m (x,t) = m> 0, and distinguish them from cases where the sourcing term, m = f(x,t) > 0, is positive but varying in space and/or time. We give a meaning and a nomenclature to the various cases and variationally characterize eingenvalue in each case. Also, there will be a distinction between various cases of boundary conditions (Drichlet, Neumann and Mixed Robin).