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

Abstract

In this paper, we are concerned with the existence of the maximum and minimum iterative solutions for a tempered fractional turbulent flow model in a porous medium with nonlocal boundary conditions. By introducing a new growth condition and developing an iterative technique, we establish new results on the existence of the maximum and minimum solutions for the considered equation; at the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also derived.

Upper and Lower Solution Method for a Singular Tempered Fractional Equation with a p-Laplacian Operator

“…On the other hand, a p-Laplacian equation can model turbulent flow in a porous medium [11][12][13][14][15]; in particular, when the equation contains tempered fractional derivatives, it can model turbulent velocity fluctuations of porous medium with features of power-law behavior at infinity and infinite divisibility [16]. Therefore, in the process of analyzing the statistical data and and modeling the basic physical phenomena in turbulent flow, Brownian motion, tempered Lévy flight, tempered stable laws are an useful tool.…”

“…•dA(t) denotes a Riemann-Stieltjes integral and A is a function of bounded variation. In our recent work [14], we studied the existence of extremal solutions for a tempered fractional turbulent flow Equation (2) in the case where γ 2 = γ 1 , δ = 1, and the nonlinearity takes the special form h(t) f (x(t)). In virtue of iterative techniques, some new results of the existence of maximum and minimum solutions were established; moreover, iterative properties of the extremal solutions such as the iterative sequences and the asymptotic estimates of solutions were also obtained.…”

Existence of positive solutions for third-order semipositone boundary value problems on time scales

“…where λ > 0 is a parameter, 1/2 < p < q < 1 are constants, f : (0, 1) × [0, ∞) → R, e : (0, 1) → R and ω : [q, 1] → [0, ∞) are continuous functions, and e ∈ L(0, 1). For more details about multiple point boundary value problems and integral boundary value problems, we refer the reader to the survey of [22,36,38] and [11,23,24,26,30,31].…”

The uniqueness and iterative properties of solutions for a general Hadamard-type singular fractional turbulent flow model

“…In comparison, the Hadamard derivative is a nonlocal fractional derivative with singular logarithmic kernel. So the study for Hadamard-type fractional differential equations is relatively difficult [9,21,27,28].…”