On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems

Abstract

In this work, we study an initial value problem for a system of nonlinear parabolic pseudo equations with Caputo fractional derivative. Here, we discuss the continuity which is related to a fractional order derivative. To overcome some of the difficulties of this problem, we need to evaluate the relevant quantities of the Mittag-Leffler function by constants independent of the derivative order. Moreover, we present an example to illustrate the theory.

Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

“…Then theory-based studies of VO calculus have been more deeply investigated by Lorenzo and Hartley [ 17 ]. Soon after, many definitions of VO derivative operators have been introduced by some researchers such as Riemann–Liouville (RL) [ 18 , 19 ], Lorenzo–Hartley [ 17 ], Coimbra [ 20 ], and Caputo [ 2 , 21 ] derivatives. These operators have been used to describe some models in a variety of science fields including biochemical tumorous bone remodeling models [ 22 ], characterizing the dynamics of van der Pol oscillators [ 23 ]; see also [ 24 , 25 ].…”

New general integral transform via Atangana–Baleanu derivatives

“…However, researchers have come to realise that models constructed from fractional calculus can successfully represent real world problems and sometimes yield better results compared to models from the integer calculus. Some useful results from fractional calculus models appear in engineering, physics, biology and economics [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

Estimates of Classes of Generalized Special Functions and Their Application in the Fractional k , s -Calculus Theory

“…In the frame of fractional derivatives, Alqahtani et al [9] studied nonlinear F-contractions on b-metric spaces and differential equations with Mittag-Leffler kernel. Many scholars have computed numerous fractional integral inequalities containing the various fractional integration and differentiation operators over the past few years (see [10,11]). The k symbols are well known from a number of sources related to the measurement of finite differences (see [12,13]).…”