On a nonlinear Volterra integrodifferential equation involving fractional derivative with Mittag-Leffler kernel

Abstract

In this paper, a nonlinear time-fractional Volterra equation with nonsingular Mittag-Leffler kernel in Hilbert spaces is studied. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, the existence of a mild solution of our problem is proved. The main tool to prove our results is the use of some Sobolev embeddings.

On maximal solution to a degenerate parabolic equation involving in time fractional derivative

“…Over the past decades, fractional calculus began to be used as a powerful tool by many researchers working in several branches of science and engineering. In many fractional models, we notice two operators of interest, namely Caputo derivative (see [11][12][13]) and Riemann-Liouville operator.…”

“…where u 0 is the function defined later. The derivative ∂ α t is the meaning of Caputo sense [11,12]. When α = 1, this above equation (1.1) is called degenerate parabolic equation which is mentioned in [4].…”

Global solutions to a time-fractional Cauchy problem

“…We refer the reader to the following works [2,3,5,8] for beatiful results of existence and uniqueness of mild solutions. In view of pseudo-parabolic models, time-fractional derivatives plays a same role as in the parabolic case.…”

Stability of a nonlinear fractional pseudo-parabolic equation system regarding fractional order of the time

“…In addition, there are numerous works on the well-posedness of the pseudo-parabolic equation with classical derivative, as evidenced by [60][61][62][63][64][65][66][67][68] and the references therein. Investigating the existence, uniqueness, and stability of fractional differential equations, has been the important goal in the scientific community, especially in fractional calculus.…”