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

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.


Introduction
Tempered stable laws were introduced to model turbulent velocity fluctuations of physics [1]. Normally, tempered stable laws retain their signature power-law behaviour at infinity and infinite divisibility [2]. By multiplying by an exponential factor for the usual second derivative, one can obtain tempered fractional derivatives and integrals. In [3], an exponential tempering factor was applied to the particle jump density in random walk and stochastic model for turbulence in the inertial range, which is the fractional derivative of Brownian motion exhibiting semilong range dependence with a power law at moderate time scales.
However, to the best of our knowledge, there are relatively few results on fractional turbulent flow in a porous medium with nonlocal Riemann-Stieltjes integral boundary conditions, and no work has been reported on the maximal and minimal solutions for the tempered-type fractional turbulent flow equation. us, following the previous work, this paper will pay attention to the extremal solutions for the tempered fractional turbulent flow equation in a porous medium with nonlocal Riemann-Stieltjes integral boundary conditions by developing iterative technique, also see [97][98][99][100]. Different from [9,83], in this paper, we will give a new type of growth condition for the nonlinear term to guarantee equation (1) has the extremal solutions. At the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also obtained.

Preliminaries and Lemmas
Before starting our work, we firstly recall the definition of the tempered fractional derivative which is an extension of the Riemann-Liouville derivative and integral.
Let λ > 0; the α-order left tempered fractional derivative is defined by where R 0 D α t denotes the standard Riemann-Liouville fractional derivative which can be found in [101]. Let e following results have been proven in [83].
has the unique solution where H(t, s) is defined by (4) and G(t, s) denotes the Green function as follows: In order to obtain the positive extremal solutions of tempered fractional turbulent flow equation (1), it is necessary to preserve nonnegativity of the Green function. (H0): where (3) where Let In order to obtain the existence of positive extremal solutions of tempered fractional turbulent flow equation (1), we introduce the following new control conditions.
(H1): f: [0, +∞) ⟶ (0, +∞) is continuous and nondecreasing, and there exists a positive constant (H2): Remark 1. Assumption (14) we introduced is a new type of growth condition, which includes a large number of basic functions such as ( Mathematical Problems in Engineering Proof. For cases (1)-(3), take respectively; obviously, For cases (4) and (5) Define a cone P, P � x ∈ E: there exists a number 0 < l x < 1 such that and an operator T in E: en, the fixed point of operator T in E is the solution of tempered fractional turbulent flow equation (1). □ Lemma 3. Assume that (H0)-(H2) hold. en, T: P ⟶ P is a continuous, compact operator.
Proof. It follows from the definition of P that, for any x ∈ P, there exists a number 0 < l x < 1 such that Since T is increasing with respect to x, by (14), (23), and Lemma 2, we have where l * x � min us, it follows from (24) that which implies that T is well defined and uniformly bounded, and T(P) ⊂ P.
On the contrary, according to the Arzela-Ascoli theorem and the Lebesgue dominated convergence theorem, it is easy to know that T: P ⟶ P is completely continuous.

Main Results
Before we begin to state our main result, we first give the following lemma.

Mathematical Problems in Engineering
By T(P δ * ) ⊂ P δ * , we have x n ∈ P δ * for n ≥ 1. It follows from the fact of T being a compact operator that x (n) is a sequentially compact set.
Letting n ⟶ + ∞, from the continuity of T and Ty (n) � y (n− 1) , we have Tx � x, which implies that x is another positive solution of equation (1). Next, we prove that x and x are the maximum and minimum positive solutions of equation (1). In fact, suppose x is any positive solution of equation (1); then, we have us, it follows from induction that Taking the limit, we have which implies that x and x are the maximal and minimal positive solutions of equation (1), respectively. In the end, since x, x ∈ P δ * ⊂ P, there exist constants n 1 > , n 2 > 0 such that

Example
Since the fractional-order derivative possesses long-memory characteristics, in fluid mechanics, equation (1) can describe a turbulent flow in a porous medium. Here, we give a specific example to illustrate the main results.
Example: consider the following nonlocal tempered fractional turbulent flow equation: where 6 Mathematical Problems in Engineering en, equation (51) has the positive minimal and maximal solutions x and x, and there exist constants n 1 > , n 2 > 0 such that Firstly, we have δ � us, (H0) holds.

Conclusion
In this work, we establish a new result on the existence of the maximum and minimum solutions for a class of tempered fractional-order differential equations with nonlocal boundary conditions. is type of equation can describe a turbulent flow of a porous medium in fluid mechanics and diffusive interaction. In order to obtain the extremal solutions of the equation, a new type of growth condition is introduced, and the iterative sequences with explicit initial values are constructed which converge uniformly to the maximum and minimum solutions; in addition, the estimations of the upper bounds of the maximum and minimum solutions are also derived.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors' Contributions
e study was carried out in collaboration among all authors. All authors read and approved the final manuscript.