On the Solution of Fractional Maxwell Equations by Sumudu Transform

In this paper, we introduce the Maxwell equations of timefractional order in lossy media. We derive the solution of these equations by using Sumudu transform techniques.


Introduction
In the literature there are numerous integral transforms and widely used in physics, astronomy as well as in engineering.In order to solve the differential equations, the integral transforms were extensively applied and thus there are several works on the theory and application of integral transforms such as the Laplace, Fourier, Mellin and Hankel, to name a few.In the sequence of these transforms, in early 90's Watugala (G.K. Watugala, 1993) introduced a new integral transform, named the Sumudu transform and further applied it to the solution of ordinary differential equation in control engineering problems.For further detail and properties about Sumudu transform (M. A. Asiru, 2001;2002;2003;F. B. M. Belgacem, 2003;M. E. El-Shandwily, 1988;H. Eltayeb, 2010;V. G. Gupta, 2010) and many others.The Sumudu transform is defined over the set of the functions: by the following formula The existence and the uniqueness was discussed in (A.Kılıc ¸man, 2010), for further properties of Sumudu transform and its derivatives we refer to (M. A. Asiru, 2001).In (M. A. Asiru, 2002), some fundamental further properties of Sumudu transform were also established.
Similarly, this new transform was applied to the one-dimensional neutron transport equation in (A. Kadem, 2005).In fact one can easily show that there is strong relationship between Sumudu and other integral transforms.In particular the relation between Sumudu transform and Laplace transforms was proved in (A.Kılıc ¸man, 2010).
Further in (H.Eltayeb, 2010), the Sumudu transform was extended to the distributions(generalized functions) and some of their properties were also studied in (A.Kılıc ¸man, 2010).Recently Kılıc ¸man et al. applied this transform to solve the system of differential equations, see (A. Kılıc ¸man, 2010).
A very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except a factor n!.Thus if n! a n t n see (J.Zhang, 2007).Similarly, the Sumudu transform sends combinations, C(m, n), into permutations, P(m, n) and hence it will be useful in the discrete systems.Further Thus we further note that since many practical engineering problems involve mechanical or electrical systems acted upon by discontinuous or impulsive forcing terms then the Sumudu transform can be effectively used to solve ordinary differential equations as well as partial differential equations in engineering problems.
The Riemann-Liouville fractional integral of order ν is defined by Miller and Ross (K.S. Miller, 1993). (3) Here we define the fractional partial-derivative for α > 0 in the form where [α] means the integral part of the number α.In particular, if 0 ≤ α < 1, (5) and if α = n, n ∈ N = {1, 2, ...}, then we have the ordinary derivative A generalization of the Mittag-Leffler function is given by and the general form as follows In order to prove our main results, we shall require the following lemma that was proved in (V.G. Gupta, 2010).
, Re α > 0 and Re β > 0 exists and given by By using the above lemma we note that Mittag-leffer function can be recovered as follows: where and where λ 1 and λ 2 are given by ( 10) and ( 11), respectively.Now taking inverse Sumudu transform and using Lemma 1, we get the desired result.

Maxwell's equations of time-fractional order and Sumudu transform
Maxewell's equations are fundamental equations in electricity and magnetism to find electromagnetic fields that can exist in any configuration (normal modes).Thus we can develop most of the working relationships in the field.Also they have successfully been applied to predict the electromagnetic fields that are produced by any distribution of charges and currents.The equations are general and no restriction is placed on the type of variation of the exciting sources.In the present paper, we introduce the Maxwell's equations of time-fractional order.Now consider the transverse electromagnetic planar where the waves propagating in the z-direction having unbounded lossy medium is given by where μ, and σ > 0 are the constants permeability, permittivity and the conductivity, respectively, see (M.E. El-Shandwily, 1988).The boundary conditions are given as follows We note that many authors have discussed the solution of ( 12) and ( 13) by using different method, for example see (M.E. El-Shandwily, 1988) and (M.G. M. Hussain, 2007).Now we consider the Maxwell's equations with time-fractional order under the same boundary conditions as follows where 0 ≤ α < 1. Taking the Sumudu transform of the above equations ( 14) and ( 15), we get Now by eliminating H(z, u) from ( 16) and ( 17), we get By applying boundary conditions, then we obtain the equation in following form By applying Sine transform to the above equation ( 19), we get Now by taking the inverse Sumudu transform and using convolution theorem, we obtain where λ 1 and λ 2 are given in ( 9) and ( 10) and a = σ / and b = s 2 μ .Now taking the inverse sine transform we obtain which is the desired solution.
For magnetic field, by using the equation ( 16), and substituting the value of ( 22), we easily get ds.
Taking the inverse Sumudu transform and by using the Lemma 1, we obtain