Periodic mild solutions of impulsive fractional evolution equations

The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations

“…. This shows that when colleges and universities expand their enrollment scale, only when the growth rate of unpaid expenses per capita for poor students is low can they consider raising education fees [6]. We have colleges and universities that charge for education as a function of the number of students enrolled.…”

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

“…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 [79][80][81][82], and tempered fractional derivatives [83]. ese works also enlarged and enriched the application of the fractional calculus in impulsive theories [84][85][86][87][88][89], chaotic system [90][91][92][93], and resonance phenomena [94][95][96]. Among them, by using the fixed point theorem of the mixed monotone operator, Zhang et al [9] established the result of uniqueness of the positive solution for the Riemann-Liouville-type turbulent flow in a porous medium:…”

Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

“…In fluid mechanics, when a fluid is subjected to a severe impact to form a fracture, singular points or singular domains also follow the fracture. Normally, at singular points and domains, the extreme behaviour such as blow-up phenomena [2,3], impulsive influence [4][5][6][7][8][9], and chaotic system [10][11][12][13], often leads to some difficulties for people in understanding and predicting the corresponding natural problems. Hence, the study of singularity for complex systems governed by differential equations [14][15][16][17][18][19][20][21][22][23][24][25][26][27] is important and interesting in deepening the understanding of the internal laws of dynamic system.…”