Fractional Derivatives in Geophysical Modelling: Approaches Using the Modified Adomian Decomposition Method
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Pure and Applied Geophysics
Fractional Derivatives in Geophysical Modelling: Approaches Using the Modified Adomian Decomposition Method JOSE´ HUMBERTO
DE
SOUZA PRATES1 and DAVIDSON MARTINS MOREIRA1
Abstract—In this paper, the modified Adomian decomposition method is utilised to solve fractional differential equations in two traditional geophysical problems: (i) the case of a current source, located on the surface of an infinite half-space, injecting electric current into the sub-surface; and (ii) the case of a plane electromagnetic wave propagating in a homogeneous and isotropic medium. The solutions obtained by the proposed method are more general than the traditional ones, in the sense the traditional results represent a special case, in which the fractional derivatives assume an integer order. The proposed methodology, which makes use of fractional modelling, is simple, easy to implement, converges fast, and enables a more accurate description of the studied phenomenon because it allows a more comprehensive investigation of the physics of the problem. Keywords: Current source, plane waves, Laplace decomposition, Adomian method, fractional differential equations.
1. Introduction Geophysics is a natural science subject that concerns the structure, composition, physical properties, and dynamics of the Earth. Apart from the study of phenomena like tsunamis, earthquakes, and volcanoes, geophysical methods are widely utilised to survey the subsurface of the Earth. A variety of methods exist, which rely on seismic, electric, electromagnetic, magnetic and gravimetric potential, radiometric and geometric measurements. These methods rely on theoretical knowledge of other subjects, such as chemistry, physics, and mathematics. This is particularly true in geophysical modelling that involves differential equations. However, the traditional differential equations presented in the literature
1 Manufacturing and Technology Integrated Campus SENAI CIMATEC, Salvador, Brazil. E-mail: davidson.moreira@ gmail.com
are usually of integer order, whereas in recent years the use of fractional derivates has been introduced, and provides a powerful tool with which to generalise differential equations (Debnath 2003; Zhang et al. 2017). In the context of traditional geophysical problems, some works have analysed the solutions of the steady-state magnetic field due to a current source in a layered Earth model (Edwards and Nabighian 1991; Chen and Oldenburg 2004). Sripanya (2011) studied the magnetic field and the electric potential considering an environment formed by layers whose conductivities are depth-dependent in exponential or binomial form. Sripanya and Yooyuanyong (2012) and Chaladgarn and Yooyuanyong (2013) obtained a solution for the magnetic field using a differential equation for a two-layer model, in which the conductivity of the first layer depends on depth in exponential form. More recently, Tunnurak et al. (2015) proposed a mathematical model relying on the finite element method to reproduce the magnetic field of an expo
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