A numerical method for solving variable-order solute transport models
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A numerical method for solving variable-order solute transport models Marjan Uddin1
· Islam Ud Din1
Received: 26 May 2020 / Revised: 9 September 2020 / Accepted: 6 October 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this work, a radial basis function (RBF)-based numerical scheme is developed which uses the Coimbra variable time fractional derivative of order 0 < α(t, x) < 1. The Coimbra derivative can efficiently model a dynamical system whose fractional-order behavior changes with time and space locations. The RBF can effectively approximate spatial derivatives in multi-dimensions. The resulted numerical scheme is RBF–FD type and is validated for solute transport problems in 1D and 2D dimension domains. Various cases of variableorder 0 < α(t, x) < 1 have been discussed. The present numerical scheme can effectively approximate those variable-order models whose exact solution can not be obtained in a simple way. Keywords RBF · Fractional order · Variable order · Coimbra derivative solute transport models · Numerical approximation Mathematics Subject Classification 65R10 · 65M12 · 35R15
Introduction In recent years, many authors have investigated that some physical systems can be best formulated by variable-order derivatives. In such like cases, the fractional-order behavior changes with time and space, respectively. For example, the time evolution of a variableorder frictional force for imposed motion in terms of well-known special functions, and a fractional or constant-order (CO) calculus model of the same problem is used as comparison. It is shown by Coimbra in (2003) that the CO-Calculus model is not able to capture all the details of the VO-Calculus solution, particularly in the areas of transition between dynamic regimes. In a similar work Ramirez and Coimbra (2010) a comparative study have been carried out using nine definitions of variable-order differential operator and Coimbra variable-order
Communicated by José Tenreiro Machado.
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Marjan Uddin [email protected] Department of Basics Sciences, University of Engineering and Technology, Peshawar, Pakistan 0123456789().: V,-vol
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derivative outperformed all the others in many aspects for modeling of a dynamical system accurately. The VO-Calculus is essential in many important physical modeling, and some excellent methods have been developed for its solutions like, solute transport problems (Kim and Kavvas 2006; Field and Leij 2012; Anderson and Phanikumar 2011; Behrens et al. 2003), visco-elasticity problems (Coimbra 2003), geographical data processing (Cooper and Cowan 2004), anomalous diffusion processes (Chen et al. 2010), signature verification processes (Tseng 2006). Some other robust methods can be found for such type of variable-order problems (see, for example, Samko 1995; Kikuchi and Negoro 1997; Ingman and Suzdalnitsky 2004; Pedro et al. 2008; Kunze et al. 2011; Anh et al. 2005; Caputo 2003; Kim and Kavvas 2006; Addison et al. 1998; Bhrawy and Zaky
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