Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

Abstract

This paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski's type. We prove that the initial value problem has the solution if and only if some initial values are zero.MSC 2010 : Primary 34A08, 34A25, 44A40; Secondary 26A33

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

“…In [12], the properties of the generalized Riemann-Liouville operator were investigated in a special functional space, and an operational method was developed for solving fractional differential equations with this operator. Based on the results of [12], the authors of [15] have developed an operational method for solving fractional differential equations containing a finite linear combination of the generalized Riemann-Liouville operators with various parameters. In [17], the problem of source identification was studied for the generalized diffusion equation with the operator D α, γ .…”

“…Note that the Laplace method was used for solving this problem in [4]. In [15], a solution was found by the operational calculus for a problem more general than (2.1) in a specially constructed functional space. In our work, in contrast to these studies, we use a more rational way to solve problem (2.1), which allows us to obtain an explicit solution.…”

“…Further, applying the operator J α 0+ to both sides of this equation and taking into account the linearity of this operator and the following formula [15]:…”

On solving fractional logistic population models with applications

“…Many real-life phenomena have been described using fractional differential equations (FDEs), such as viscoelasticy (Koeller 1984), continuum mechanics (Carpinteri and Mainardi 1997), optimal control (Bhrawy and Ezz-Eldien 2016), hydrologic modeling (Benson et al 2013), variational problems (Ezz-Eldien 2016), fluid mechanics (Kulish and Lage 2002), finance (Jiang et al 2012), and others (Gaul et al 1991;Ferreira et al 2008; Ezz-Eldien and El-Kalaawy 2018; Dzielinski et al 2010). Searching for numerical techniques for approximating the solutions of FDEs has been strongly considered in the last decades, such as the radial basis functions method (Hosseini et al 2014), the Haar wavelet method , the fractional finite volume method (Liu et al 2014), the Adomian decomposition method (Khodabakhshi et al 2014), the operational method Kim et al 2014), and so on (Heydari et al 2013;Bhrawy and Zaky 2017;Ezz-Eldien and Doha 2018;Stern et al 2014). As a spectral approach for solving numerically some kinds of differential equations, the tau method is classified as one of the most important used methods due to its high convergence and accuracy.…”

Modified Galerkin algorithm for solving multitype fractional differential equations

“…In addition, we apply the new algorithm for solving fractional partial differen- [3][4][5][6][7][8][9][10][11][12][13] The development of efficient numerical techniques for approximating the solutions of fractional differential equations (FDEs) has been an important issue in the last decades. Therefore, strategies such as the fractional finite volume, 14 Haar wavelet, 15 Adomian decomposition method, 16 radial basis functions, 17 operational matrix 18 methods, and other approaches [19][20][21][22][23] were proposed.…”