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

Authors: Xinguang Zhang; Jiqiang Jiang; Lishan Liu; Yonghong Wu
Journal: Mathematical Problems in Engineering
 (2020)

---

## 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.


---

## Body

## 1. 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 semi-long range dependence with a power law at moderate time scales.Tempered stable laws are useful in statistical physics and provide a basic physical model such as turbulent flow for the underlying physical phenomena. Motivated by these physical backgrounds and the sources, in this paper, we focus on the existence of the maximum and minimum iterative solutions for the following tempered fractional turbulent flow equation with nonlocal boundary conditions:(1)D0Rtα,λφpD0Rtβ,λxt=htfxt,x0=x′0=⋅⋅⋅=xn−20=D0Rtβ,λx0=0,D0Rtβ,λx1=∫01e−λ1−txtdt,D0Rtγ,λφpD0Rtβ,λx1=∫01atD0Rγ,λφpD0Rtβ,λxtdAt,where 1<α≤2,n−1<β≤n,n≥4,0<γ<α−1, λ>0 is a constant, φp is the p-Laplacian operator defined by φps=sp−2s,p>1, D0Rtα,λ is the tempered fractional derivative, ∫01atD0Rtγ,λφpD0Rtβ,λxtdAt denotes a Riemann–Stieltjes integral, A is a function of bounded variation, f:0,+∞⟶0,+∞ is continuous, and h∈L0,1,0,+∞.Turbulent flow is a fundamental fluid mechanics problem which can be described by ap-Laplacian equation with a suitable boundary condition; for details, see [4]. Particularly, if the model is of fractional order, then it can describe turbulent flow in a porous medium [5–10]. On the contrary, fractional-order derivative has nonlocal characteristics; based on this property, the fractional differential equation can also interpret many abnormal phenomena that occur in applied science and engineering, such as viscoelastic dynamical phenomena [11–29], advection-dispersion process in anomalous diffusion [30–34], and bioprocesses with genetic attribute [35, 36]. 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, 37–62], Hadamard-type derivatives [63–71], Katugampola–Caputo derivatives [72], conformable derivatives [73–76], Caputo–Fabrizio derivatives [77, 78], Hilfer derivatives [79–82], and tempered fractional derivatives [83]. These works also enlarged and enriched the application of the fractional calculus in impulsive theories [84–89], chaotic system [90–93], and resonance phenomena [94–96]. 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:(2)Dtβφp−Dtαzt=−fzt,Dtγzt,t∈0,1,Dtαz0=Dtα+1z0=Dtαz1=0,Dtγz0=0,Dtγz1=∫01DtγzsdAs,where 0<γ≤1<α≤2<β<3,α−γ>1, Dtα,Dtβ,andDtγ denote the Riemann–Liouville derivatives, and ∫01zsdAs indicates the Riemann–Stieltjes integral, and A is a function of bounded variation; the nonlinear term may be singular at both first variable and second variable. Recently, Zhou et al. [83] investigated a class of tempered fractional differential equations with Riemann–Stieltjes integral boundary conditions; by using the fixed point theorem of the sum-type mixed monotone operator, the existence and uniqueness of positive solutions were established, and iterative sequences for approximating the unique positive solution were also constructed.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. Thus, 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 [97–100]. 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.


## 2. 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.Letλ>0; the α-order left tempered fractional derivative is defined by(3)D0Rtα,λxt=e−λtD0Rtαeλtxt,where D0Rtα denotes the standard Riemann–Liouville fractional derivative which can be found in [101].Let(4)Ht,s=β1−sβ−1β−1+seλstβ−1−ββ−1eλst−sβ−1e−λtβ−1Γβ+1,0≤s≤t≤1,β1−sβ−1β−1+seλsβ−1Γβ+1tβ−1e−λt,0≤t≤s≤1,Δ=e−λ−δΓα−γ,δ=∫01e−λssα−γ−1asdAs.The following results have been proven in [83].Lemma 1.
Givenk∈C0,1; then, the boundary value problem,(5)D0Rtα,λφpD0Rtβ,λxt=kt,x0=x′0=⋅⋅⋅=xn−20=D0Rtβ,λx0=0,D0Rtβ,λx1=∫01e−λ1−txtdt,D0Rtγ,λφpD0Rtβ,λx1=∫01atD0Rtγ,λφpD0Rtβ,λxtdAt,has the unique solution(6)xt=∫01Ht,sφq∫01Gt,τkτdτds,where Ht,s is defined by (4) and Gt,s denotes the Green function as follows:(7)Gt,s=G1t,s+tα−1e−λtΔΓα−γ∫01atG2t,sdAt,G1t,s=eλs−tΓα1−sα−γ−1tα−1−t−sα−1,0≤s≤t≤1,1−sα−2tα−1,0≤t≤s≤1,G2t,s=eλs−tΓα1−sα−γ−1tα−γ−1−t−sα−γ−1,0≤s≤t≤1,1−sα−γ−1tα−γ−1,0≤t≤s≤1.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:(8)∫01e−λssα−γ−1asdAs<e−λ.Lemma 2.
AssumeH0 holds; then, functions Gt,s and Ht,s have the following properties:(1)
Gt,s and Ht,s are nonnegative and continuous for t,s∈0,1×0,1.(2)
For anyt,s∈0,1,Ht,s satisfies(9)m1se−λttβ−1≤Ht,s≤M1se−λttβ−1,where(10)M1s=β1−sβ−1β−1+seλsβ−1Γβ+1,m1s=βs1−sβ−1eλsβ−1Γβ+1.(3)(11)m2se−λttα−1≤Gt,s≤M2se−λttα−1,s,t∈0,1,where(12)M2s=1Γα+δΔΓαΓα−γeλs1−sα−γ−1,m2s=eλs1−sα−γ−1−1−sα−1Γα.Let(13)M1∗=β2eλβ−1Γβ+1.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 ϵ>3/q−1 such that(14)0<d≕sups≥0fss+2ϵ<+∞.H2:(15)0<∫01M2τhτdτ<M1∗3ϵq−11/q−1d−1.Remark 1.
Assumption (14) we introduced is a new type of growth condition, which includes a large number of basic functions such as(1)
ft=b0+∑i=1nbit+2μi, where b0>0,bi,μi>3/q−1,i=1,2,…,n.(2)
ft=b0+∑i=1nbit+2μi1/μ, where b0>0,μ>0,bi>0,μi>0i=1,2,…,n and(16)mini=1,…,nμi>3μq−1.(3)
ft=t+1μ−1ln1+1/2+t+2+tμ−2+b,b>0,μ>2+3/q−1.(4)
f:0,+∞⟶0,+∞ is continuous and nondecreasing, and there exists a positive constant ϵ>3/q−1 such that fx/x+2ϵ is increasing on x, and(17)limx⟶+∞fxx+2ϵ=M>0.(5)
f:0,+∞⟶0,+∞ is continuous and nondecreasing, and there exists a positive constant ϵ>3/q−1 such that fx/x+2ϵ is nonincreasing on x.Proof.
For cases (1)–(3), take(18)ϵ=maxi=1,2,…,nμi,ϵ=maxi=1,2,…,nμiμ,ϵ=μ−2,respectively; obviously,(19)0<d≕sups≥0fss+2ϵ<+∞.
For cases (4) and (5), it is clear; we omit the proof.
DenoteE=C0,1 as all continuous functions equipped the maximum norm(20)x=maxxt:t∈0,1.
Define a coneP,(21)P=x∈E:there exists a number0<lx<1such that0≤xt≤lx−1e−λttβ−1,t∈0,1,and an operator T in E:(22)Txt=∫01Ht,sφq∫01Gs,τhτfxτdτds.
Then, the fixed point of operatorT in E is the solution of tempered fractional turbulent flow equation (1).Lemma 3.
Assume thatH0–H2 hold. Then, T:P⟶P is a continuous, compact operator.Proof.
It follows from the definition ofP that, for any x∈P, there exists a number 0<lx<1 such that(23)0≤xt≤lx−1e−λttβ−1,t∈0,1.
SinceT is increasing with respect to x, by (14), (23), and Lemma 2, we have(24)Txt=∫01Ht,sφq∫01Gs,τhτfxτdτds≤e−λttβ−1∫01M1sφq∫01Gs,τhτfxτdτds≤e−λttβ−1∫01M1sφq∫01Gs,τhτfxτxτ+2ϵxτ+2ϵdτds≤φqde−λttβ−1∫01M1sφq∫01Gs,τhτxτ+2ϵdτds≤φqde−λttβ−1∫01M1sφq∫01Gs,τhτlx−1e−λττβ−1+2ϵdτds≤M1∗φqdlx−1+2ϵ∫01M2τhτdτe−λttβ−1≤1lx∗e−λttβ−1,where(25)lx∗=min12,M1∗−1dlx−1+2ϵ∫01M2τhτdτ1−q.
Thus, it follows from (24) that(26)0≤Txt≤1lx∗e−λttβ−1,which implies that T is well defined and uniformly bounded, and TP⊂P.
On the contrary, according to the Arzela–Ascoli theorem and the Lebesgue dominated convergence theorem, it is easy to know thatT:P⟶P is completely continuous.


## 3. Main Results

Before we begin to state our main result, we first give the following lemma.Lemma 4.
Supposeϵq−1>3 and H2 hold; then, the equation(27)M1∗d∫01M2τhτdτq−1x+3ϵq−1x+1−1=1,has unique solution δ∗ in 0,∞.Proof.
Let(28)ϕx=1−M1∗d∫01M2τhτdτq−1x+3ϵq−1x+1−1.
It follows fromϵq−1>3 and H2 that(29)ϕ0=1−M1∗d∫01M2τhτdτq−13ϵq−1>0,(30)ϕ+∞=−∞.
On the contrary,ϕx is a continuous function in 0,∞ satisfying(31)ϕ′x=−ϵq−1M1∗d∫01M2τhτdτq−1x+3ϵq−1−1x+1−1+M1∗d∫01M2τhτdτq−1x+3ϵq−1x+1−2=−M1∗d∫01M2τhτdτq−1x+3ϵq−1−1x+1−1ϵq−1−x+3x+1<0,x∈0,∞.
Thus, (29)–(31) imply equation (27) has unique zero point δ∗in 0,∞.Theorem 1.
SupposeH0–H2 hold. Then, the following is obtained:(i)
Existence: equation (1) has a positive minimal solution x¯ and a positive maximal solution x¯.(ii)
Asymptotic estimates: there exist positive numbersni>0,i=1,2, such that(32)x¯te−λttβ−1∈0,n1,x¯te−λttβ−1∈0,n2,t∈0,1.(iii)
Iterative sequences: for initial valuesx0t=0andy0t=δ∗+1, construct the iterative sequences(33)xnt=∫01Ht,sφq∫01Gs,τhτfxn−1τdτds,ynt=∫01Ht,sφq∫01Gs,τhτfyn−1τdτds.Then,(34)limn⟶+∞xnt=x¯t,limn⟶+∞ynt=x¯t,uniformly, for t∈0,1, where δ∗ is the unique solution of equation (27) in 0,∞.Proof.
Firstly, letPδ∗=x∈P:0≤x≤δ∗+1; we shall show TPδ∗⊂Pδ∗.
For anyx∈Pδ∗ and for any t∈0,1, we have(35)0≤xt≤maxt∈0,1xt≤δ∗+1.
Consequently, it follows fromH1 and Lemma 4 that(36)Tx=maxt∈0,1∫01Ht,sφq∫01Gs,τhτfxτdτds≤∫01M1sφq∫01Gs,τhτfxτdτds≤∫01M1sφq∫01Gs,τhτfxτxτ+2ϵxτ+2ϵdτds≤φqd∫01M1sφq∫01Gs,τhτxτ+2ϵdτds≤φqd∫01M1sφq∫01Gs,τhτδ∗+3ϵdτds≤M1∗d∫01M2τhτdτq−1δ∗+3ϵq−1=δ∗+1,which implies that TPδ∗⊂Pδ∗.
Next, take the initial valuex0t=0, and let(37)x1t=Tx0t=T0t,t∈0,1.
It follows fromx0t∈Pδ∗ that x1t∈TPδ∗⊂Pδ∗.
Denote(38)xn+1t=Txnt=Tn+1x0t,n=1,2,….
ByTPδ∗⊂Pδ∗, we have xn∈Pδ∗ for n≥1. It follows from the fact of T being a compact operator that xn is a sequentially compact set.
On the contrary, sincex1t≥0=x0t and T is increasing on x, we have(39)x2t=Tx1t≥Tx0t=x1t,t∈0,1.
By induction, one has(40)0≤xnt≤xn+1t≤δ∗+1,n=1,2,….
Consequently, there existsx¯∈Pδ∗ such that xn⟶x¯. Noticing that Txn=xn−1 and letting n⟶+∞, by the continuity of T, we have Tx¯=x¯, which implies that x¯ is a nonnegative solution of equation (1), and then x¯ is a positive solution of equation (1) since f0>0.
Now, we takey0t=δ∗+1 as the initial value and let(41)y1t=Ty0t,t∈0,1.
It follows fromy0t=δ∗+1∈Pδ∗ that y1∈Pδ∗. Thus, construct the iterative sequence(42)yn+1t=Tynt=Tn+1y0t,n=1,2,….
We have(43)ynt∈Pδ∗,n=0,1,2,…,since TPδ∗⊂Pδ∗. It follows from Lemma 3 that yn is a sequentially compact set.
Now, sincey1∈Pδ∗ and T is increasing, one has(44)0≤y1t≤y1≤δ∗+1=y0t,and then(45)y2t=Ty1t≤Ty0t=y1t.
It follows from induction that(46)0≤yn+1t≤ynt≤δ∗+1,n=0,1,2,…,which implies that there exists x¯∈Pδ∗ such that yn⟶x¯. Letting n⟶+∞, from the continuity of T and Tyn=yn−1, we have Tx¯=x¯, which implies that x¯ is another positive solution of equation (1).
Next, we prove thatx¯ 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(47)x0t=0≤x˜t≤δ∗+1=y0t,x1t=Tx0t≤Tx˜t=x˜t≤Ty0t=y1t.
Thus, it follows from induction that(48)xnt≤x˜t≤ynt,n=1,2,3,….
Taking the limit, we have(49)x¯≤x˜≤x¯,which implies that x¯ and x¯ are the maximal and minimal positive solutions of equation (1), respectively.
In the end, sincex¯,x¯∈Pδ∗⊂P, there exist constants n1>,n2>0 such that(50)x¯te−λttβ−1∈0,n1,x¯te−λttβ−1∈0,n2,t∈0,1.


## 4. 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:(51)D0Rt3/2,1φ3/2D0Rt7/2,1xt=e−t1−t−1/4400xt+13ln1+12+xt+2+xt2+2,x0=x′0=x′′0=D0Rt7/2,1x0=0,D0Rt7/2,1x1=∫01e−1−txtdt,D0Rt1/4,1φ3/2D0Rt7/2,1x1=∫01D0Rt1/4,1φ3/2D0Rt7/2,1xtdAt,where(52)At=0,t∈0,12,1,t∈12,34,12,t∈34,1.Then, equation (51) has the positive minimal and maximal solutions x¯ and x¯, and there exist constants n1>,n2>0 such that(53)x¯te−tt5/2∈0,n1,x¯te−tt5/2∈0,n2,t∈0,1.Let(54)α=32,β=72,λ=1,γ=14,p=32,at=1,ht=1400e−t1−t−1/4,fx=x+13ln1+12+x+2+x2+2.Firstly, we have(55)δ=∫01e−λssα−γ−1asdAs=∫01e−ss1/4dAs=0.2902<e−1=0.3679.Thus,H0 holds.Obviously,f:0,+∞⟶0,+∞ is continuous and nondecreasing. Take ϵ=2>3/q−1=3/2; then, we have(56)0<d≕supx≥0fxx+2ϵ=supx≥0x+1x+23ln1+12+xx+2+1+2x+22=2<+∞,which implies that H1 is satisfied.Now, we computeM1∗andΔ:(57)M1∗=β2eλβ−1Γβ+1=3.52×e2.5×Γ4.5=0.1550,Δ=e−λ−δΓα−γ=e−1−0.3679Γ5/4=0.0857.Thus, we have(58)M2s=1Γ3/2+0.36790.0857×Γ3/2Γ5/4es1−s1/4=5.3439es1−s1/4.Consequently,(59)0<∫01M2τhτdτ=0.013358<M1∗3ϵq−11/q−1d−1=0.0219.So, conditionH3 holds.Thus, by Theorem1, equation (51) has a positive minimal solution x¯ and a positive maximal solution x¯, and there exist constants n1>,n2>0 such that(60)x¯te−tt5/2∈0,n1,x¯te−tt5/2∈0,n2,t∈0,1.


## 5. 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. This 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.



---

*Source: 2492193-2020-09-25.xml*