Representation of solution for a linear fractional delay differential equation of Hadamard type

Abstract

This paper is devoted to seeking the representation of solutions to a linear fractional delay differential equation of Hadamard type. By introducing the Mittag-Leffler delay matrix functions with logarithmic functions and analyzing their properties, we derive the representation of solutions via the constant variation method.

Positive Solutions for a Weakly Singular Hadamard-Type Fractional Differential Equation with Changing-Sign Nonlinearity

“…Thus, (27), (28), Lemma 1, (F1), and the Arzela-Ascoli theorem guarantee that the operator T : B → P is completely continuous.…”

“…In particular, the Hadamard derivative is a nonlocal fractional derivative with singular logarithmic kernel. So the study of Hadamard fractional differential equations is relatively difficult; see [26][27][28][29][30][31][32].…”

Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

“…Our definition of the two-parameter delayed M-L type matrix function with logarithm differs substantially from the definition given in [33].…”

“…However, we find that there exists only one [33] work on the representation of explicit solutions of Hadamard type fractional order delay linear differential equations. In [33] authors studied the Hadamard type fractional linear time-delay system…”

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

“…On the contrary, fractional-order derivative has nonlocal characteristics; based on this property, the fractional differential equation can also interpret many abnormal phenomena that occur in applied science and engineering, such as viscoelastic dynamical phenomena [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], advection-dispersion process in anomalous diffusion [30][31][32][33][34], and bioprocesses with genetic attribute [35,36]. As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann-Liouville derivatives [16,, Hadamardtype derivatives [63][64][65][66][67][68][69][70][71], Katugampola-Caputo derivatives [72], conformable derivatives [73][74][75][76], Caputo-Fabrizio derivatives [77,78], Hilfer derivatives …”