Evolution equations with a parameter and application to transport-convection differential equations

Abstract

We deeply investigate the well-posedness of models taking the formt is a derivative with the fractional parameter β and A is a closed densely defined operator in a Banach space. We show that, unlike other systems, solutions of our models are not governed by Mittag-Leffler functions and their variants. We extend and adapt Peano's idea to our models and establish conditions for existence and uniqueness of solutions. In particular, relations between the two-parameter solution operator, its resolvent, and its generator are provided; the issue of subordination and prolongation principles are addressed; and a way to approximate the generalized solution is presented. Finally, application to transport-convection differential equations is performed in the space of distributions with finite higher moments to show how their well-posedness can be addressed.

Existence Results for a Class of Semilinear Fractional Partial Differential Equations with Delay in Banach Spaces

“…By general Banach contraction mapping principle, for operator Q there exists a unique fixed point ∈ ([− , ]; ), which means that ∈ ([− , ]; ) is the unique mild solution of problem (12). That is, problem (3) has a unique mild solution.…”

“…Definition (see [24]). If ∈ ([− , 0 ]; ) is a mild solution of problem (12), then satisfies the following integral equations:…”

“…In [7,10], the authors studied the existence results of the fractional integrodifferential equations of order 1 < ≤ 2. In [11,12], the authors considered a class of fractional differential equations, where the fractional derivative operator is 0 with fractional order and is a closed densely defined operator in a Banach space. Goufo [13] studied the existence results for a class of fractional fragmentation model by theory of strongly continuous solution operators.…”

Heat Transfer on Fractionalized Micropolar Nanofluid over Oscillating Plate via Caputo-Fabrizio Fractional Operator

“…Finally, expressing the governing equations (25)- (27) in terms of Caputo-Fabirizio fractional derivative, we have:…”

“…Haq et al [13] explored the e ects of magnetic eld with water functionalized metallic nanoparticles for squeezed ow over a sensor surface. In short, this study includes few latest references on nano uids [14][15][16], modern fractional derivatives [17][18][19][20][21][22], heat transfer [23][24][25][26][27], nanoparticles [28][29][30][31], porosity and magnetic eld [32][33][34][35][36][37], and few di erent circumstances [38][39][40][41][42][43]. Motivated by the above research work on nano uids, the purpose of this study is to investigate the analytic and mathematical performance of micropolar nano uid to enhance heat transfer.…”