A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations

Solution of Fractional Optimal Control Problems by using orthogonal collocation and Multi-objective Optimization Stochastic Fractal Search

“…These mathematical tools are used to approximate the term d μ f (t)/dt μ (μ is a fractional order in a fractional differential problem) found in FOCP. The most commonly used approaches are presented in Podlubny (1999), Demirci and Ozalp (2012), Rehman and Khan (2012), Aslefallah and Rostamy (2014), Liu and Hou (2017), Yang et al (2017a), Yang et al (2017b), Li and Rui (2018), Liang et al (2018), Zhang (2018), and Yang et al (2019).…”

“…The study of fractional calculus arises by the end of the seventeenth century. In the specialized literature, various contributions can be found, such as those on chemistry (Kirchner et al 2000), finance (Sabatelli et al 2002), hydrology (Schumer et al 2003), biology (Magin 2006), viscoelasticity (Larsson et al 2015), chaos synchronization (Su et al 2016), robotics (Kumar and Rana 2017), anomalous diffusion (Zhang et al 2017), anomalous heat-diffusion problems (Yang et al 2017a), advection-dispersion problems (Yuan et al 2016;Zhang, 2018;Li and Rui 2018), fractional Boussinesq equation (Yang et al 2017b), diffusion equations (Yang et al 2018), anomalous advection-dispersion equations (Liang et al 2018), solution of direct and inverse fractional advection-dispersion problem (Lobato et al 2019), and anomalous diffusion equations (Yang et al 2019).…”

Solution of mass-spring-damper fractional systems using Caputo derivative and orthogonal collocation

“…In this section, notations and definitions used to evaluate the derivative of a generic function f with respect to time t (d μ f(t)/dt μ , where μ is a fractional order in a FDE) are presented (Podlubny, 1999;Demirci and Ozalp, 2012;Rehman and Khan, 2012;Aslefallah and Rostamy, 2014;Liu and Hou, 2017;Yang et al, 2017aYang et al, , b, 2019Li and Rui, 2018;Liang et al, 2018;Zhang, 2018).…”

“…In the past decades, different applications considering models represented by fractional order in different fields of science have been published in the literature. Among them, we can cite studies on chemistry (Kirchner et al , 2000), finance (Sabatelli et al , 2002; Bohner and Hatipoğlu, 2018, 2019), hydrology (Schumer et al , 2003), biology (Magin, 2006), viscoelasticity (Larsson et al , 2015), eigenvalue problem (Jin and Liu, 2016), robotics (Kumar and Rana, 2017), anomalous diffusion (Zhang et al , 2017), anomalous heat-diffusion problems (Yang et al , 2017a), advection-dispersion problems (Yuan et al , 2016; Zhang, 2018; Li and Rui, 2018), fractional Boussinesq equation (Yang et al , 2017b), diffusion equations (Yang et al , 2018; Shi et al , 2020), chaos synchronization (Su et al , 2018), anomalous advection-dispersion equations (Liang et al , 2018), inverse fractional advection-dispersion problem (Lobato et al , 2019), anomalous diffusion equations (Yang et al , 2019), risk theory (Constantinescu et al , 2019), HIV dynamics (Wasques et al , 2020), solution of mathematical equations using spline collocation methods (Zahra et al , 2020) and Langevin equation (Yang et al , 2020).…”