Analytical and Numerical Treatment of Maxwell's Equations

Maxwell's equations are one of the most important models in different fields. It describes electromagnetic phenomena such as micro, radios and radar waves. The modeling of systems involving electromagnetic waves is widely spread and has attracted the attention of many authors and researchers. In this work, we will present some important analytical and numerical aspects of Maxwell's equations. We will review some basic properties of electromagnetic theory, namely: electromagnetic fields, magneto-static fields, and time varying fields. Moreover, we will use these physical properties to derive Maxwell's equations in various forms. Though, it is well known that Maxwell's equations are hard to solve analytically, however, we will attempt to use some well known analytical methods to solve these equations in some particular domains such as a sphere and a circular cylinder. Such analytical methods include: separation of variables, series expansion method, conformal mapping and integral methods such as Laplace transforms and cosine and sine Fourier transforms. Numerical methods for solving Maxwell's equations are extensively used nowadays and are usually referred to as Computational Electro-magnetic (CEM). Here the Finite Difference and Finite Difference Time Domain Method (FDTDM) known for its simplicity and efficiency will be proposed to solve Maxwell's equations. And the Yee Algorithm will also be illustrated. Moreover, the convergence, stability and error analysis for these numerical methods will also be investigated.