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ORIGINAL RESEARCH article

Front. Phys., 23 November 2020
Sec. Statistical and Computational Physics
This article is part of the Research Topic New Trends in Fractional Differential Equations with Real-World Applications in Physics View all 16 articles

A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

\nHari Mohan SrivastavaHari Mohan Srivastava1Pshtiwan Othman Mohammed
Pshtiwan Othman Mohammed2*
  • 1Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
  • 2Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah, Iraq

We consider the comparison theorems for the fractional forward h-difference equations in the context of discrete fractional calculus. Moreover, we consider the existence and uniqueness theorem for the uncertain fractional forward h-difference equations. After that the relations between the solutions for the uncertain fractional forward h-difference equations with symmetrical uncertain variables and their α-paths are established and verified using the comparison theorems and existence and uniqueness theorem. Finally, two examples are provided to illustrate the relationship between the solutions.

1. Introduction

The study of fractional calculus and fractional differential equations has received recent attention from both applied and theoretical disciplines. Indeed, it was observed that the use of them are very useful for modeling many problems in mathematical analysis, medical labs, engineering sciences, and integral inequalities (see for e.g., [114]). There is much interesting research on what is usually called integer-order difference equations (see for e.g., [15, 16]). Discrete fractional calculus and fractional difference equations represent a new branch of fractional calculus and fractional differential equations, respectively. Also, for scientists, they represent new areas that have, in their early stages, developed slowly. Some works are dedicated to boundary value problems, initial value problems, chaos, and stability for the fractional difference equations (see for e.g., [1723]).

Besides the discrete fractional calculus, the uncertain fractional differential and difference equations have been introduced and investigated in order to model the continuous or discrete systems with memory effects and human uncertainty (see for e.g., [2428]). In Lu and Zhu [27], the relations between uncertain fractional differential equations and the associated fractional differential equations have been created via comparison theorems for fractional differential equations of Caputo type in Lu and Zhu [26]. Lu et al. [28] presented analytic solutions to a type of special linear uncertain fractional difference equation (UFDE) by the Picard iteration method. Moreover, they provided an existence and uniqueness theorem for the solutions by applying the Banach contraction mapping theorem. After that, Mohammed [29] generalized the above work.

Nowadays, discrete fractional calculus shows incredible performance in the fields of physical and mathematical modeling. The motivation behind solving the fractional difference equations relies on fast investigation of the properties within models of fractional sum and difference operators (see for e.g., [20, 3036]).

Motivated by the aforementioned results, we will try to create a link between uncertain fractional forward h-difference equations (UFFhDEs) and associated fractional forward h-difference equations (FFhDEs) in the sense of Riemann–Liouville fractional operators via the comparison theorems and existence and uniqueness theorem.

The rest of our article is designed as follows. In section 2, we presented the preliminary definitions and important features that are useful in the accomplishment of this study. In section 3, the comparison theorems of the fractional differences are pointed out. Inverse uncertainty distribution, the existence and uniqueness theorem, the relation between UFFhDEs and associated FFhDEs, and some related examples are pointed out in section 4. Finally, the future scope and concluding remarks are summarized in section 5.

2. Preliminaries

In what follows, we recall some results in discrete fractional calculus that has been developed in the last few years; for more details, we refer to references [2428, 28, 29, 37, 38] and the related references therein.

Definition 2.1 ([39]). The forward difference operator on hℤ is defined by

Δhf(η)=f(η+h)-f(η)h,

and the backward difference operator on hℤ is defined by

hf(η)=f(η)-f(η-h)h.

For h = 1, we get the classical forward and backward difference operators Δψ(η) = ψ(η + 1) − ψ(η) and ∇ψ(η) = ψ(η) − ψ(η − h), respectively. The forward jumping operator on hℤ is σ(r) = r + h and the backward jumping operator is ρ(r) = rh.

For a, b ∈ ℝ with a<b,b-ah and 0 < h ≤ 1, we use the notations ℕa,h = {a, a + h, a + 2h, ...}, b,hℕ = {b, bh, b − 2h, ...}.

Definition 2.2 ([39]). Let η, θ ∈ ℝ and 0 < h ≤ 1, the delta h-factorial of η is defined by

ηh(θ)=Γ(ηh+1)Γ(ηh+1-θ),    (2.1)

where we use the convention that division at a pole yields zero and θ is the falling delta h-factorial order of η. It is worth mentioning that ηh(θ) is a function of η for given θ and h.

Definition 2.3 ([37, 38, 40]). Let f be defined on ℕa,h for the left case and b,hℕ for the right case. Then, the left delta h-fractional sum of order θ > 0 is defined by

(Δahθψ)(η)=aσ(ηθ h)(ησ(τ))h(θ1)ψ(τ)Δhτ                        =1Γ(θ)r=ahηhθ(ησ(rh))h(θ1)ψ(rh)h,     ηa+θ h,h,

and the right delta h-fractional sum is defined by

(Δhbθψ)(η)=ρ(η+θ h)b(ρ(τ)η)h(θ1)ψ(τ)hτ                              =1Γ(θ)r=ηh+θbh(rhσ(η))h(θ1)ψ(rh)h,                              ηbθ h,h.

Lemma 2.1 ([40]). Let θ, μ > 0, h > 0, and p be defined on ∈ ℕa,h. We then have

(Δa+μhhθΔahμp)(η)=(Δah(μ+θ)p)(η)                                                      =(Δa+θhhμΔahθp)(η),    (2.2)

for all η ∈ ℕa + (θ + μ)h,h.

Lemma 2.2 ([40]). Let θ > 0 and ψ be defined ona,h and b,hℕ, respectively. Then the left and right delta h-fractional differences of order θ are defined by

(Δahμψ)(η)=(ΔhmΔah(mμ)ψ)(η),    (2.3)
(Δhbμψ)(η)=(1)m(hmΔhb(mμ)ψ)(η),    (2.4)

where m = [θ] + 1.

Lemma 2.3 ([40]). Let ψ be defined ona,h, then, for any θ > 0, we have

(ΔahθΔhψ)(η)=ΔhΔahθψ(η)(ηa)h(θ1)Γ(θ)ψ(a).    (2.5)

Lemma 2.4 ([40]). Let θ > 0, μ > 0, and h > 0, and we then have

Δa+μhhθ(ηa)h(μ)=Γ(μ+1)Γ(μ+θ+1)(ηa)h(θ+μ),Δhbμhθ(bη)h(θ)=Γ(μ+1)Γ(μ+θ+1)(bη)h(θ+μ).

Lemma 2.5 ([40]). Let θ ∈ ℝ and q be any positive integer, then

(ΔahθΔhqψ)(η)=(ΔhqΔahθψ)(η)                                      k=0q1(ηa)h(vq+k)Γ(vq+k+1)Δhkψ(a),    (2.6)

for η ∈ ℕa + θ h,h.

Lemma 2.6 ([38]). Suppose that μh,μh+θ\{...,-2,-1}, then we have

Δahθ(ηa+μ)h(μh)=Γ(μh+1)Γ(μh+θ+1)(ηa+μ)h(μh+θ),

for each η ∈ ℕah,h.

Lemma 2.7. Let ψ be defined ona,h and m be a positive integer with 0 < m − 1 < μm. The definition of the fractional h-difference (2.3) is then equivalent to

(Δahμψ)(η)={1Γ(μ)r=ahηh+μ(ησ(rh))h(μ1)ψ(rh)h,m1<μ<m,Δahmp(η),μ=m,

for η ∈ ℕa,h.

Motivated by the definition of nth order forward sum for uncertain sequence ξη, we define the θth order forward sum for uncertain sequence ξη as follows:

Definition 2.4. Let θ be a positive real number, a ∈ ℝ, and ξη be an uncertain sequence indexed by η ∈ ℕa,h. Then,

Δahθξη=1Γ(θ)r=ahηhθ(ησ(rh))h(θ1)ξrhh

is called the θth order forward fractional sum of uncertain sequence ξη, where σ(r) = r + h.

Definition 2.5. The fractional Riemann–Liouville-like forward difference for uncertain sequence ξη is defined by

Δahμξη=Δhn(Δah(nμ)ξη),

where θ > 0 and 0 ≤ n − 1 < μ ≤ n, n represents a positive integer.

3. The Comparison Theorems

Consider the following FFhDEs:

Δ(θn)hhθψ(η)=g(η+(θn)h,ψ(η+(θn)h)),    (3.1)

subject to the initial conditions

Δ(θn)hhθn+iψ(η)|t=0=ψi,     i=0,1,,n1,    (3.2)

where (θ-n)hΔhθ denotes a fractional Riemann–Liouville forward h-difference with 0 ≤ n − 1 < θ ≤ n, g is a real-valued function defined on [0, ∞) × ℝ, η ∈ ℕ0,h, and ψi ∈ ℝ for i = 0, 1, ..., n − 1.

Now, by applying the operator 0Δh-θ to Equation (3.1), then the initial value problem (3.1) and (3.2) is equivalent to the following fractional sum equation:

ψ(η)=i=0n-1(η)h(θ-n+i)Γ(θ-n+i+1)ψi          +1Γ(θ)r=0ηh-θ(η-σ(rh))h(θ-1)g(r+(θ-n)h,ψ(r+(θ-n)h))h,    (3.3)

where we have used Lemma 2.1, Lemma 2.5, and the fact that ΔhnΔh-nψ(η)=ψ(η).

First, a comparison theorem for Riemann–Liouville fractional h-difference equations with θ ∈ (0, 1] will be presented.

Theorem 3.1. Suppose g(η, ψ) and k(η, ψ) are two real-value functions defined on [0, ∞] × ℝ. Function k is Lipschitz continuous in y with Lipschitz constant Lk that has 0<Lkh-θθ. If ψ1(η) and ψ2(η) are, respectively, unique solutions of the following IVPs

{Δ(θ1)hhθψ(η)=g(η+(θ1)h,ψ(η+(θ1)h)),     η0,Δ(θ1)hhθ1ψ(η)|t=0=X0,    (3.4)

and

{Δ(θ1)hhθψ(η)=k(η+(θ1)h,ψ(η+(θ1)h)),     η0,Δ(θ1)hhθ1ψ(η)|t=0=ψ0.    (3.5)

1. if g(η, ψ) ≤ k(η, ψ), then ψ1(η) ≤ ψ2(η) for each η ∈ ℕ(θ − 1)h,h,

2. if g(η, ψ) > k(η, ψ), then ψ1(η) > ψ2(η) for each η ∈ ℕθ h,h.

Proof: (1) Assume that the condition ψ1(η) ≤ ψ2(η) is not valid; there thus exists η0 ∈ ℕ(θ−1)h,h such that ψ1(η0) > ψ2(η0). Let η1 = min{η ∈ ℕ(θ−1)h,h; ψ1(η) > ψ2(η)} and X(η) = ψ1(η) − ψ2(η). Then, we have

X(η1)>0,    (3.6)
X(η)0,     η(θ-1)h,h[0,η1-h].    (3.7)

Considering the fractional sum equations equivalent to IVPs (3.4) and (3.5), we have

ψ1(θ h)=θ hv-1ψ0+hθg((θ-1)h,X0),ψ2(θ h)=θ hv-1ψ0+hθk((θ-1)h,X0).

Subtracting these and then making use of hθ > 0 for h > 0, θ ∈ (0, 1], and g(η, ψ) ≤ k(η, ψ), we get

ψ1(θ h)-ψ2(θ h)=hv(g((θ-1)h,X0)-k((θ-1)h,X0))0.

This verifies that η1 > θ h. From this and since η1 ∈ ℕ(θ − 1)h,h, we can write η1 = (θ + ℓ)h, l = 1, 2, .... By Lemma 2.6, we then get

(θ1)hΔhθX(η1θh)                        =1Γ(θ)r=θ1η1h(η1θhσ(rh))h(θ1)X(rh)h                        =1Γ(θ)r=θ1θ+(hσ(rh))h(θ1)X(rh)h                        =hθX((θ+)h)θhθX((θ+1)h)                        +1Γ(θ)r=θ1θ+2(hσ(rh))h(θ1)X(rh)h.

That is,

h-θX((θ+)h)=(θ-1)hΔhθX(η1-θ h)+θh-θX((θ+-1)h)                                 -1Γ(-θ)r=θ-1θ+-2( h-σ(rh))h(-θ-1)X(rh)h.    (3.8)

Now, by using the Lipschitz continuity of k in y, g(η, x) ≤ k(η, x), and (3.7), we get

Δ(θ1)hhθX(η1θh)=Δ(θ1)hhθψ1(η1θh)                                                 Δ(θ1)hhθψ2(η1θh)                                                 =g(η1h,ψ1(η1h))                                                 k(η1h,ψ2(η1h))                                                 k(η1h,ψ1(η1h))                                                  k(η1h,ψ2(η1h))                                                  Lk(ψ1(η1h)ψ2(η1h))                                                  LkX(η1h).

Denoting ω(η1-h):=(θ-1)hΔhθX(η1-θ h)+LkX(η1-h), it follows that

ω((θ+-1)h)0.    (3.9)

This gives

Δ(θ1)hhθX(η1θh)=LkX((θ+1)h)+ω((θ+1)h).

Thus, Equation (3.8) becomes

h-θX((θ+)h)=(θ h-θ-Lk)X((θ+-1)h)+ω((θ+-1)h)                                 -1Γ(-θ)r=θ-1θ+-2( h-σ(rh))h(-θ-1)X(rh)h.    (3.10)

We write r = v − 1 + i, i = 0, 1, ..., ℓ − 1 to obtain

( h-σ(rh))h(-θ-1)Γ(-θ)=( h-(θ+i)h)h(-θ-1)Γ(-θ)            =h-θ-1Γ(-i+1-θ)Γ(-θ)Γ(-i+2)            =h-θ-1(-i-θ)(-i-1-θ)(-θ)Γ(-θ)Γ(-θ)Γ(-i+2)            =h-θ-1(-θ)(-θ+1)(-i-1-θ)(-i-θ)Γ(-i+2)            =h-θ-1(-θ)(-θ+1)(-θ-1+c)(-θ+c)Γ(c+1),             where c=-i.

Since θ ∈ (0, 1] and h−θ−1 > 0, so

( h-σ(rh))h(-θ-1)Γ(-θ)0.    (3.11)

Considering Lk<θh-θ, h−θ > 0 and Equations (3.9)–(3.11), it follows that

h-θX((θ+)h)0.

This implies that X(η1) ≤ 0, which contradicts with (3.6).

(2) By the same technique of (1), we assume that the condition ψ1(η) > ψ2(η) is not valid. There thus exists η2 ∈ ℕθh,h, such that ψ1(η2) ≤ ψ2(η2). Let η3 = min {η ∈ ℕθh,h; ψ1(η) ≤ ψ2(η)} and z(η) = ψ2(η) − ψ1(η). We then have

z(η3)0,    (3.12)
z(η)<0,     ηθ h,h[0,η3-h].    (3.13)

Considering the fractional sum equations equivalent to IVPs (3.4) and (3.5), hθ > 0 and g(η, ψ) > k(η, ψ), we find ψ1h) > ψ2h). That is; η3 > θ h. If we write η3 = (θ + ℓ)h, l = 1, 2, ..., then, by Lemma 2.6, we get

Δ(θ1)hhθz(η3θh)                  =1Γ(θ)r=θ1η3h(η3θhσ(rh))h(θ1)z(rh)h                  =hθz((θ+)h)θhθz((θ+1)h)                  +1Γ(θ)r=θ1θ+2(hσ(rh))h(θ1)z(rh)h,

or equivalently,

h-θz((θ+)h)=(θ-1)hΔhθz(η3-θ h)+θ h-θz((θ+-1)h)                                -1Γ(-θ)r=θ-1θ+-2( h-σ(rh))h(-θ-1)z(rh)h.    (3.14)

Now, by using the Lipschitz continuity of k in y, g(η, z) > k(η, z), and (3.13), we get

Δ(θ1)hhθz(η3θh)=Δ(θ1)hhθψ1(η3θh)                                                Δ(θ1)hhθψ2(η3θh)                                                =w(η3h,ψ2(η3h))                                                k(η3h,ψ1(η3h))                                                k(η3h,ψ2(η3h))                                                 k(η3h,ψ1(η3h))                                                 Lk(ψ2(η3h)ψ1(η3h))                                                 Lkz(η3h).

Denoting w(η3-h):=(θ-1)hΔhθz(η3-θ h)+Lkz(η3-h), it follows that

w((θ+-1)h)0.    (3.15)

This gives

Δ(θ1)hhθz(η3θh)=Lkz((θ+1)h)+w((θ+1)h).

Equation (3.14) thus becomes

h-θz((θ+)h)=(θh-θ-Lk)z((θ+-1)h)+w((θ+-1)h)                                -1Γ(-θ)r=θ-1θ+-2( h-σ(rh))h(-θ-1)z(rh)h.    (3.16)

Similarly for θ ∈ (0, 1] and h−θ−1 > 0, we can show that

( h-σ(rh))h(-θ-1)Γ(-θ)0.    (3.17)

Considering Lk<θ h-θ, h−θ > 0 and Equations (3.15)–(3.17), it follows that

h-θz((θ+)h)0.

This implies that z(η3) ≤ 0, which contradicts with (3.12). The proof of Theorem 3.1 is thus completed.

In the sequel, we will extend a comparison theorem for Riemann-Liouville fractional h-difference equations of the order θ with 0 ≤ n − 1 < θ ≤ n.

Theorem 3.2. Suppose g(η, ψ), and k(η, ψ) are two real-value functions defined on [0, ∞] × ℝ. Function k is Lipschitz continuous in y with a Lipschitz constant Lk that has 0<Lkh-θθ. If ψ1(η) and ψ2(η) are, respectively, unique solutions of the following IVPs

{ (θn)hΔhθψ(η)=g(η+(θn)h,ψ(η+(θn)h)),     η0,    (θn)hΔhθn+iψ(η)|t=0=ψi,  i=0,1,  , n1    (3.18)

and

{ (θn)hΔhθψ(η)=k(η+(θn)h,ψ(η+(θn)h)),     η0, (θn)hΔhθn+iψ(η)|t=0=ψi,  i=0,1,,  n1.    (3.19)

1. if g(η, ψ) ≤ k(η, ψ), then ψ1(η) ≤ ψ2(η) for each η ∈ ℕ(θ−n)h,h,

2. if g(η, ψ) > k(η, ψ), then ψ1(η) > ψ2(η) for each η(θ-n+1)hh.

Proof: (1) For μ = θ − n + 1 ∈ (0, 1] and η ∈ ℕ0,h, we have Δ(θn)hhθψ(η)=Δhn1Δ(μ1)hhμψ(η). By using Lemma 2.5, the IVPs (3.18) and (3.19) can be easily converted to the following IVPs, respectively,

{ (μ1)hΔhμψ(η)=1Γ(n1)r=0ηh(n1)(ησ(rh))h(μ2)g(r+(μ1)h,                                  ψ(r+(μ1)h))h+i=0n2(η)h(i)Γ(i+1)ψi+1, (μ1)hΔhμ1ψ(η)|t=0=ψ0,    (3.20)

and

{ (μ1)hΔhμψ(η)=1Γ(n1)r=0ηh(n1)(ησ(rh))h(μ2)k(r+(μ1)h,                                ψ(r+(μ1)h))h+i=0n2(η)h(i)Γ(i+1)ψi+1, (μ1)hΔhμ1ψ(η)|t=0=ψ0.    (3.21)

Denote

g¯(η,x)=1Γ(n1)r=0ηh(n1)(ησ(rh))h(n2)g(r+(μ1)h,ψ(r+(μ1)h))h+i=0n2(η)h(i)Γ(i+1) ψi+1,

and

k¯(η,x)=1Γ(n1)r=0ηh(n1)(ησ(rh))h(n2)k(r+(μ1)h,ψ(r+(μ1)h))h+i=0n2(η)h(i)Γ(i+1) ψi+1,

for η ∈ ℕ(θ−1)h,h. These give

g¯(η,x)k¯(η,x)=1Γ(n1)r=0ηh(n1)(ησ(rh))h(n2)               ×  [g(r+(μ1)h,ψ(r+(μ1)h))k(r+(μ1)h,             ψ(r+(μ1)h))] h.    (3.22)

Since g(η, ψ) ≤ k(η, ψ) and

(η-σ(rh))h(n-2)Γ(n-1)=(η-(r+1)h)h(n-2)Γ(n-1)                                      =h-n-2Γ(ηh-r)Γ(n-1)Γ(ηh-r-n+2)                                      =h-n-2Γ(c)Γ(n-1)Γ(c-n+2),                                            where c=ηh-r,r=0,1,...,ηh-n+1>0,

it follows from (3.22) that g¯(η,ψ)k̄(η,ψ) for η ∈ ℕ(θ−1)h,h. Then, by applying Theorem 3.1 for the above findings, we get ψ1(η) ≤ ψ2(η) for η ∈ ℕ(θ−n)h,h. Hence, the proof of the first item is completed.

(2) Analogously, we can obtain the proof of this item, and thus our proof is completely done.

4. Inverse Uncertainty Distribution

In this section, we make a link between the solution for an UFFhDE and the solution for the associated FFhDE; we firstly define a symmetrical uncertain variable and α-path for an UFFhDE in view of Lu and Zhu [27]. After that, we state and verify a theorem that demonstrates a link between solution for the UFFhDE with symmetrical uncertain variables and its α-path via the comparison theorems in section 3. To understand the theory of inverse uncertainty distribution, we advise the readers to read [41] carefully.

First, we recall the inverse uncertainty distribution theory:

Definition 4.1 ([41]). An uncertainty distribution Ψ is called regular if it is a continues and strictly increasing function and satisfies

limx-Ψ(x)=0,     limx+Ψ(x)=1.    (4.1)

Definition 4.2 ([41]). Let ξ be an uncertain variable with a regular uncertainty distribution Ψ. Then, the inverse function Ψ−1 is called the inverse uncertainty distribution of ξ.

Example 4.1. From Definition 4.2, we deduce that

(i) the inverse uncertainty distribution of a linear uncertain variable L(a,b) is given by

Ψ-1(θ)=(1-θ)a+θ b;    (4.2)

(ii) the inverse uncertainty distribution of a normal uncertain variable N(e,σ) is given by

Ψ-1(θ)=e+3σπln(θ1-θ);    (4.3)

(iii) and the inverse uncertainty distribution of a normal uncertain variable LOGN(e,σ) is given by

Ψ-1(θ)=exp(e)+(θ1-θ)3σπ.    (4.4)

Definition 4.3 ([41]). We say that an uncertain variable ξ is symmetrical if

Ψ(x)+Ψ(-x)=1,    (4.5)

where Ψ(x) is a regular uncertainty distribution of ξ.

Remark 4.1. From definition 4.3, we can deduce that the symmetrical uncertain variable has the inverse uncertainty distribution Ψ−1(θ), which satiates

Ψ-1(θ)+Ψ-1(1-θ)=0.    (4.6)

Example 4.2. From definition 4.3, we deduce the following:

1. the linear uncertain variable L(-a,a) is symmetrical for any positive real number a.

2. The normal uncertain variable N(0,1) is symmetrical.

Consider the following UFFhDE with Riemann-Liouville-like forward difference:

(θn) hΔhθX(η)=F(η+(θn)h,X(η+(θn)h))                                    +G(η+(θn)h,X(η+(θn)h))ξη+(θn)h,    (4.7)

subject to the crisp initial conditions

(θn)hΔhθnkX(η)|t=0=Xk,     k=0,1,,n1,    (4.8)

where (θ-n)hΔhθ denotes a fractional Riemann–Liouville forward h-difference with 0 ≤ n − 1 < θ ≤ n, M, N are two real-valued functions defined on [0, ∞) × ℝ, η ∈ ℕ0,h ∩ [0, Th], Xk ∈ ℝ for k = 0, 1, ..., n − 1, and ξ(θ − n)h, ξ(θ−n+1)h, ⋯, ξη+(θ−n)h are i.i.d. uncertain variables with symmetrical uncertainty distribution L(a,b).

Definition 4.4 ([41]). An UFFhDE (4.7) with crisp initial conditions (4.8) is said to have an α-path if it is the solution of the corresponding FFhDE

Δ(θn)hhθX(η)=F(η+(θn)h,X(η+(θn)h))                                    +|G(η+(θn)h,X(η+(θn)h))|Ψ1(θ)    (4.9)

with the same initial conditions (4.8), where Ψ−1(θ) is the inverse uncertainty distribution of uncertain variables ξη for η ∈ ℕ(θ−n)h,h ∩ [0, Th].

Theorem 4.1. Let η ∈ ℕ0,h ∩ [0, Th], n ∈ ℕ, λ ∈ (0, 1) and θ ∈ (0, 1]. The linear UFFhDE:

Δ(θn)hhθX(η)=λX(η+(θn)h)+λξη+(θn)h,

with the initial conditions

(θn)hΔhθniX(η)|t=0=Xi,     i=0,1,,  n1,

has a solution

X(η)=XiFμ,λ;h(η)+ξη,     ,i=0,1,...,n-1,

where ξη is an uncertain sequence with the uncertainty distribution L(a·eθ,λ;h(η),b·eθ,λ;h(η)), and

Fθ,λ;h(η)=k=0λki=0n-1(η+k(θ-n)h)h((k+1)θ h-nh+i)Γ((k+1)θ-n+i+1),

and

eθ,λ;h(η)=k=1λk(η+(k-1)(θ-n)h)h(kθ)Γ(kθ+1).

Proof: By making the use of Lemma 2.5, we can easily prove this theorem by the similar technique of [29, Theorem 3.1], so it is omitted.

Example 4.3. Consider the following UFFhDE:

(θ1)hΔhθX(η)=λX(η+(θ1)h)+λξη+(θ1)h,                                  η0,h[0,Th],λ(0,1),θ(0,1],    (4.10)

where ξ(θ−1)h, ξθ h, …, ξη+(θ−1)h are i.i.d linear uncertain variable L(-2,2), which has the inverse uncertainty distribution Ψ−1(θ) = 4θ − 2 by (4.2).

By Theorem 4.1, the associated FFhDE of (4.10) with its initial condition

                 (θ1)h ΔhθX(η)=λX(η+(θ1)h)+λΨ1(θ),(θ1)hΔhθ1X(η)|t=0=X0

has a solution

X(η)=X0k=0λk(η+k(θ-1)h)h((k+1)θ-1)Γ((k+1)θ)          +k=1λk(η+(k-1)(θ-1)h)h(kθ)Γ(kθ+1)(4θ-2).

The UFFhDE (4.10) has an α-path

Xηθ=X0k=0λk(η+k(θ-1)h)h((k+1)θ-1)Γ((k+1)θ)       +k=1λk(η+(k-1)(θ-1)h)h(kθ)Γ(kθ+1)(4θ-2).

with the initial condition (θ-1)hΔhθ-1X(η)|t=0=X0.

Example 4.4. Consider the following UFFhDE:

(θ2)hΔhθX(η)  =q X(η+(θ2)h)+qξη+(θ1)h,                                        η0,h[0,Th],q(0,1),θ(0,1],    (4.11)

where ξ(θ−2)h, ξ(θ−1)h, …, andξη+(θ−2)h are the i.i.d normal uncertain variable N(0,1), which has the inverse uncertainty distribution Ψ-1(θ)=3πln(θ1-θ) by (4.2).

By Theorem 4.1, the associated FFhDE of (4.11) with its initial condition

                         Δ(θ2)hhθX(η)=qX(η+(θ2)h)+qΨ1(θ),Δ(θ2)hhθ2+iXi(η)|t=0=Xi,     i=0,1

has a solution

X(η)=k=0qki=01Xi(η+k(θ-2)h)h((k+1)θ h-2h+i)Γ((k+1)θ-1+i)          +3πln(θ1-θ)k=1qk(η+(k-1)(θ-2)h)h(kθ)Γ(kθ+1).

The UFFhDE (4.11) has an α-path

Xηθ=k=0qki=01Xi(η+k(θ-2)h)h((k+1)θ h-2h+i)Γ((k+1)θ-1+i)     +3πln(θ1-θ)k=1qk(η+(k-1)(θ-2)h)h(kθ)Γ(kθ+1).

with the initial condition (θ-2)hΔhθ-2+iXi(η)|t=0=Xi,     i=0,1.

In the following theorem, we make a relationship between uncertain fractional forward h-difference equations (UFFhDEs) and fractional h-difference equations (FFhDEs) based on the comparison theorems in section 3.

Theorem 4.2. If Xη and Xηθ are the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively. Assume that F + |G−1(θ) is a Lipschitz continues function in x with a Lipschitz constant Lk that has 0<Lk<θ h-θ. Assume that ξη is the i.i.d. symmetrical uncertain variable for η(θ-(n-1))h,hh[0,Th], then

(i) XηXηθ if ξη(γ)Ψ-1(θ) for ηD+ and ξη(γ)Ψ-1(1-θ) for ηD-, where

D+={η(θ-(n-1))h,h[0,Th];  G(η,x)0},

and

D-={η(θ-(n-1))h,h[0,Th];  G(η,x)<0},

(ii) Xη>Xηθ if ξη(γ)>Ψ-1(θ) for ηD+ and ξη(γ)<Ψ-1(1-θ) for ηD-.

Proof: First, we let ξη(γ)Ψ-1(θ) for ηD+. Then η ∈ ℕ(θ−(n−1))h,h ∩ [0, Th] and G(η, x) ≥ 0. Therefore,

G(η,x)ξη(γ)|G(η,x)|Ψ-1(θ).    (4.12)

Moreover, if ξη(γ)Ψ-1(1-θ) for ηD-, we have η ∈ ℕ(θ−(n−1))h,h ∩ [0, Th] and G(η, x) < 0. Since ξη is symmetrical, we have Ψ−1(θ) + Ψ−1(1 − θ) = 0. Thus,

G(η,x)ξη(γ)G(η,x)Ψ-1(1-θ)=-G(η,x)Ψ-1(θ)                          =|G(η,x)|Ψ-1(θ).    (4.13)

Since Xη(γ) and Xηθ are the unique solution and α-path of UFFhDE (4.7) with the initial conditions (4.8), respectively, we have

Δ(θn)hhθX(η)=F(η+(θn)h,X(η+(θn)h))                                    +G(η+(θn)h,X(η+(θn)h))ξη+(θn)h(γ),    (4.14)
Δ(θn)hhθX(η)=F(η+(θn)h,X(η+(θn)h))                                    +|G(η+(θn)h,X(η+(θn)h))|Ψ1(θ).    (4.15)

Hence, by use of Theorem 3.2 with (4.12)–(4.15), we get the proof of item (i). The proof of the second item (ii) is similar to (i). Thus, the proof of Theorem 4.2 is completed.

Theorem 4.3 (Existence and Uniqueness). Assume that F(η, x) and G(η, x) satisfy the Lipschitz condition

|F(η,x)-F(η,ψ)|+|G(η,x)-G(η,ψ)|L|x-y|,    (4.16)

and there is a positive number L that satisfies the following inequality:

L<h-θ-1Γ(θ+1)Γ(T+1-θ)Γ(T+1)(Q+1),    (4.17)

where Q = |a| ∨ |b|. Then UFFhDE (4.7) with the initial conditions (4.8) has a unique solution X(η) for η ∈ ℕθh,h ∩ [0, Th].

Proof: Proof of this theorem is similar to the existence and uniqueness theorem [29, Theorem 3.2], and it is therefore omitted.

Example 4.5. Consider the following UFFhDE:

1Δ20.5X(η)=sin X(η1)50+(η1)2+ξη1,     η02[0,8],    (4.18)

where ξ−1, ξ1, ξ3, ξ5, ξ7 are 5 i.i.d. linear uncertain variables with linear uncertainty distribution L(-2,2).

In this example h = 2, θ = 0.5, T = 4,

|F(η,x)-F(η,ψ)|+|G(η,x)-G(η,ψ)|150|x-y|=0.02|x-y|,

and

h-θ-1Γ(θ+1)Γ(T+1-θ)Γ(T+1)(Q+1)=2-1.5Γ(0.5+1)Γ(4+1-0.5)3Γ(4+1)                                                             0.05>0.02.

Thus, the existence and uniqueness Theorem 4.3 confirms that UFFhDE (4.18) has a unique solution.

Now, since

F(η,x)+|G(η,x)|Ψ-1(θ)=sin x50+(η-1)2+4θ-2,

we deduce that F(η, x) + |G(η, x)|Ψ−1(θ) is Lipschitz continues in x with Lipschitz constant L = 0.02 < 0.35 = θ h−θ.

We see that G(η, x) = 1 > 0, and, from example 4.2, we see L(-2,2) is symmetrical. Hence, by Theorem 4.2, we deduce the following link between unique solution and α-path of UFFhDE (4.18):

(i) XηXηθ if ξη ≤ 4θ − 2,

(ii) Xη>Xηθ if ξη > 4θ − 2.

Example 4.6. Consider the following UFFhDE:

Δ381214X(η)=0.025X2(η38)+ξη38,     η012[0,32],    (4.19)

where ξ-38,ξ18,ξ58,ξ98 are 4 i.i.d. linear uncertain variables with linear uncertainty distribution L(-3,3).

In this example h = 0.5, θ = 0.25, T = 3,

|F(η,x)-F(η,ψ)|+|G(η,x)-G(η,ψ)|0.025|x+y||x-y|                                                         =0.1|x-y|,     for x[-2,2],

and

h-θ-1Γ(θ+1)Γ(T+1-θ)Γ(T+1)(Q+1)=(12)-54Γ(0.25+1)Γ(3+1-0.25)4Γ(3+1)                                                             0.4>0.1.

Thus, the existence and uniqueness Theorem 4.3 confirms that UFFhDE (4.19) has a unique solution.

Now, since

F(η,x)+|G(η,x)|Ψ-1(θ)=0.025x2+6θ-3,

we deduce that F(η, x) + |G(η, x)|Ψ−1(θ) is Lipschitz, continued in x with Lipschitz constant L = 0.1 < 0.3 = θ h−θ.

We see that G(η, x) = 1 > 0, and, from example 4.2, we see L(-3,3) is symmetrical. Hence, by use of Theorem 4.2, we deduce that XηXηθ if ξη ≤ 6θ − 3 and Xη>Xηθ if ξη > 6θ − 3. This is a link between unique solution and α-path of UFFhDE (4.19).

5. Conclusions

We have considered the fractional forward h-difference equations and uncertain fractional forward h-difference equations in the context of discrete fractional calculus. The comparison theorems and existence and uniqueness theorem for the FFhDEs and UFFhDEs have been found. From a theoretical point of view, we have created a strong relationship between the solutions for UFFhDEs with the symmetrical uncertain variables and the solutions for associated UFFhDEs (namely the α-path of UFFhDEs).

Our presented results are in the sense of Riemann-Liouville fractional operator. It is important to point out the future scope of our results. There is an important task here that the researchers will be able to consider in the future. What is the task? The interested readers can extend the ideas that were presented in this article to the two well-known models of fractional calculus that were defined by operators similar to the Riemann-Liouville fractional operator but with Mittag-Leffler functions in the kernel, namely the Atangana–Baleanu (or briefly AB) [42, 43] and Prabhakar [44] models.

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: uncertain fractional h-difference equations, the comparison theorems, α-paths, existence and uniqueness theorem, discrete fractional calculus

Citation: Srivastava HM and Mohammed PO (2020) A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths. Front. Phys. 8:280. doi: 10.3389/fphy.2020.00280

Received: 09 March 2020; Accepted: 22 June 2020;
Published: 23 November 2020.

Edited by:

Jordan Yankov Hristov, University of Chemical Technology and Metallurgy, Bulgaria

Reviewed by:

Praveen Agarwal, Anand International College of Engineering, India
Amar Debbouche, 8 May 1945 University of Guelma, Algeria

Copyright © 2020 Srivastava and Mohammed. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Pshtiwan Othman Mohammed, pshtiwansangawi@gmail.com

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