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1. Introduction

In recent years, many researchers were interested in the study of turbulent flow in which the fluid undergoes irregular fluctuations or mixing. For example, Leibenson [17] intro- duced a Laplacian differential equation , to model turbulent flow in a porous medium, where 1. However, the transport of solute in highly heterogeneous porous media often exhibits anomalous diffusion phe- nomenon [14], and laboratory data and numerical experiments [2, 8] have indicated that solutes moving through a highly heterogeneous porous media violate the basic Fick’s first law of Brownian motion. Thus the works [2, 8] indicated that the fractional differential equation is more suitable for describing the convection–dispersion process of solutes in porous media.

On the other hand, since fractional-order derivative possesses a nonlocal character- istics, so it can provide a possibility to represent the memory occurring in viscoelas- tic dynamical process [4, 10, 13], blood flow [3], quantum mechanics [7], advection– dispersion process in anomalous diffusion [22, 23, 25] and bioprocesses with genetic attribute [5, 20, 26]. As a powerful tool of modeling the above many abnormal phe- nomena, in the last few decades, the fractional calculus theory has been enriched, and several different derivatives and integrals such as Caputo, Atangana, Riemann–Liouville, Hadamard, Caputo–Fabrizio, Hilfer, Riesz derivative and so on have been developed. 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].

Thus, in this paper, we choose a general Hadamard-type singular fractional differential equation involving a nonlinear operator from turbulent flow to study, more specifically, we consider the iterative properties of positive solutions for the equation

where are and order Hadamard fractional derivatives, is a differential operator denoted by , that is, is a continuous function with singularities at , 1 and ., and is a nonlinear operator satisfying

which possesses the following properties:

Proposition 1.

(i) has a nonnegative increasing inverse mapping ;

(ii) ;

(iii) ;

Equation (1) involves a nonlinear operator , which implies that Eq. (1) covers many interesting and important cases, in particular, if 1 and then , in this case, Eq. (1) reduces to the following form:

where are the and order Hadamard fractional derivatives, is a differential operator denoted by . By using the fixed point index and the properties of nonnegative matrices Ding et al. [9] considered the existence of positive solutions for a system of the above Hadamard-type fractional differential equations with semipositone nonlinearities. If 1, Eq. (1) takes the form

which is a p-Poisson turbulent flow equation in highly heterogeneous porous media. By using the fixed point theorem of the mixed monotone operator Zhang et al. [22] studied the uniqueness of positive solution for a fractional-order model of turbulent flow in a porous medium. In addition, on fractional differential equations, some significant work by using fixed theorems has been made by Karapinar and his collaborators [1, 11, 12]. Thus Eq. (1) is a more generalized p-Poisson turbulent flow equation in highly heterogeneous porous media (3). To the best of our knowledge, no results have been reported on the iterative analysis of positive solutions for Eq. (1) involving a nonlinear operator under singular case.

In addition, a fluid in highly heterogeneous porous media may push the transmission process from a phase into another different phase or state. At absolute zero, this change always leads to transformation process losing continuity and further forms some singular points or singular domains. Normally, near singular points and domains, the “bad” prop- erties such as blow-up properties [24], impulse interference [19], chaotic influence [6] obstruct people’s apperceiving for the essence of related natural phenomena. Thus it is important and interesting to explore the properties of dynamic process governed by singular differential equations, which can deepen people’s comprehension for the natural law of dynamic system.

Thus, to overcome the difficult associated with singular logarithmic kernel and singu- lar nonlinearity of Eq. (1) and follow the work Nieto [18] and Ladde, Lakshmikantham and Vatsala [16], a new double monotone iterative technique will be developed, and more new estimates are given. This paper is organized as follows. In Section 2, we firstly give the definition of Hadamard fractional integral and differential operators and then claim the properties of Green function. The main results are summarized in Section 3.

2. Preliminaries and lemmas

In this section, we firstly review the definitions of the Hadamard-type fractional integrals and derivatives; for details, see [15].

and let be a finite or infinite interval of .

The order left Hadamard fractional integral is defined by

and the α-order left Hadamard fractional derivative is defined by

In the following, we firstly consider the linear Hadamard fractional equation

Equation (4) is equivalent to the following Hammerstein-type integral equation [9]:

and

is the Green’s function of Eq. (4).

Let

for , and then we consider the associated linear boundary value problem

We have the following lemma.

Lemma 1.The associated linear boundary value problem (6) has a unique solution if and only if it solves the Hammerstein-type integral equation

Proof. Let then the associated linear boundary value problem (6) reduces to

By (4) it is equivalent to the Hammerstein-type integral equation

Noting that as well as (5) and Proposition 1, the associated linear boundary value problem (6) is equivalent to the following equation:

and it follows from [9] that if and only if it solves the following Hammerstein-type integral equation

the proof is completed.

Lemma 2. . Then the Green’s functions H and G has the following properties:

ecuacion

(i) H, G.

(ii) For all t, s, the following inequalities hold:

Our work space of this paper is Banach space E = C[1, e] equipped with the norm . Let , then P is a normal cone of E with normality constant 1. Now let us define a subset of P by

lu(ln t)β−1 ≤ u(x) ≤ lu−1(ln t)β−1, x ∈ [1, e]}

and a nonlinear operator T : E → E:

It follows from Lemma 1 that is a solution of the Hadamard-type fractional turbulent flow model (1) if and only if is a fixed point of the nonlinear operator T .

3. Main results

In order to carry out the iterative analysis of positive solution for the singular Hadamard- type fractional turbulent flow equation (1), we need the following assumption:

(H1) is continuous and decreasing with regard to second variable, and for any , there exists a constant 0 , where p is defined by (2), such that for all,

Remark 1. Assumption (7) allows f to have a singularity at 1, e, and 0. For example, satisfies assumption (H1).

Remark 2. Suppose that condition (H1) holds. It is easy to prove that for any 1 and for all , the following formula is also valid:

Now we give our main results as follows:

Theorem 1. Suppose that(H1)holds, and the following condition is satisfied:

Then

(i) The singular Hadamard-type fractional turbulent flow equation (1) has a unique positive solution u^∗ ∈ Λ;

(ii) For any initial value v₀ ∈ Λ, we construct the iterative sequence

then converges uniformly to the unique positive solution u^∗ of Eq. (1) on [1, e];

(iii) The error of the iterative value v_n and the exact solution u^∗ can be estimated by the following formula:

with an exact convergence rate

where 0 < ζ < 1 is a positive constant, which is determined by and some ρ ∈ (0, 1);

(iv) The exact solution u^∗ of Eq. (1) has an entire asymptotic estimate

Proof. Firstly, we show that T : Λ → Λ is a compact operator.

To do this, for any Λ, by the definition of the set Λ there exists a constant 0 < l_u < 1 such that

Notice that is decreasing in , it follows from Lemma 2, (7), (9) and Proposition 1 that

So T is well defined and uniformly bounded.

On the other hand, notice that His uniformly continuous on [1, e] × [1, e], and let 1 for all u ∈ Λ. Then we have

that is, T (Λ) is equicontinuous, and then T is a compact operator in Λ.

Now we show that T (Λ) ⊂ Λ. In fact, similar to (10), for any u ∈ Λ, one has

and

where satisfies

Hence we have from (11) and (12).

Next, by developing the double monotone iterative technique we will show that Eq. (1) has a unique positive solution u^∗ in Λ.

Taking , we have Λ, and then Λ, which implies that there exists a constant 0 1 such that

where is chosen according to (13). Notice that 0 < 1 for some . (0, 1) and take a sufficiently large positive constant such that

Now we construct an iterative sequence with initial value

Then we assert

In fact, noticing that T is a decreasing operator in ., by (14)–(16) we have

and then

On the other hand, from (7) and (14) we have the following estimates:

By using (8), (14), (20) and the monotonicity of . we get

Thus it follows from (18), (19) and (21) that

So by mathematical induction the conclusion (17) holds.

On the other hand, for any (0, 1), it follows from Proposition 1, (7) and (8) that

Since is increasing operator with respect to ., thus let us apply (22) repeatedly, one gets

that is,

So for any natural numbers and , we have

and

Notice that P is a normal cone with normality constant 1, thus it follows from (23), (24) that

which implies that is a Cauchy sequence of compact set. Since {un} ∈ Λ and is compact, consequently, converges to some with

and then

Let in (25), one has , that is, is a positive solution of Eq. (1). In the end, we show the uniqueness of in and the asymptotic properties of solution for Eq. (1). Suppose that is another positive solution . Let . Clearly, 0 . We claim that 1. Otherwise, we have 0 , which as well as (22) implies

Since , this contradicts with the definition of So we have 1, and then . In the same way, we have . Consequently, , so the positive solution of Eq. (1) in . is unique.

On the other hand, for any initial value , there exists a constant (0, 1) such that

Let

Since there still exists a constant l (0, 1) such that

As (15), let us choose a sufficiently large 0 such that

where (0, 1). Thus

which implies that , and then

According to the fact that is increasing operator with respect to and (26), we have

Letting in (27), we get that uniformly converges to the unique positive solution of Eq. (1).

Moreover it follows from (24) and (27) that the unique positive solution satisfies the following estimate of error:

with a exact convergence rate

where 0 1 is a positive constant, which is determined by .

Finally, since , we have the following entire asymptotic analysis, that is, there exists a constant 0 < l < 1 such that

The proof is completed.

4. Example

The following example shows that conditions of Theorem 1 are easily verified.

Consider the following singular Hadamard-type fractional turbulent flow model:

with a nonlinear operator .

Then we have the following conclusions:

(i) The singular Hadamard-type fractional turbulent flow equation (28) has a unique positive solution ;

(ii) For any initial value , we construct the iterative sequence

then converges uniformly to the unique positive solution of Eq. (28) on [1, e];

(iii)The error of the iterative value and the exact solution can be estimated by the following formula:

and the convergence rate is

where 0 < ζ < 1 is a positive constant;

(iv) The exact solution of Eq. (28) has an entire asymptotic estimate

First, we define a set as

there exists a positive number 0 1 such that

then we have

and

Clearly, is continuous and decreasing with respect to variable, and for any (0, 1), there exists a constant 0 1 such that for all ,

Moreover, we also have

and then

Thus all conditions of Theorem 1 are satisfied. From Theorem 1 it follows that all of the conclusions hold.

Now we give the graphical illustration of the iterative process to show the effectiveness of approximate solution converging to the exact solution.

Figure 1 shows that the convergence speed of iterative sequence is fairly fast, espe- cially when . = 4, the iterative solution has almost approximated exact solution of Eq. (28). Tables 1 and 2 give a numerical approximation to exact solution, which also shows that iterative process is very effective and the iterative convergence speed is robust.

5. Conclusion

In this paper, we study the uniqueness and iterative properties of solutions for a general Hadamard-type singular fractional turbulent flow model involving a nonlinear operator. The singularities and nonlinear operator lead to some difficulties of study. To overcome these difficulties, we introduce a new double monotone iterative technique and give some useful estimations. Then we establish some new results of positive solutions including the uniqueness, the iterative sequence converging to the unique solution, error estimates, convergence rate and entire asymptotic behavior. These properties can describe nicely the dynamic behaviour of the turbulent flow. The example also indicates that the conditions of theorem are easy to be verified, the graphic and numerical approximation show that the convergence speed of iterative sequence is robust and effective.

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Notes

Notas de autor

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