On Fuzzy Conformable Double Laplace Transform with Applications to Partial Differential Equations

1  Introduction

1.1 Research Background

A natural way to model uncertainty is through fuzzy differential equations [1,2], and [3]. Having these models and solutions requires an understanding of the dynamics of design [4]. The entire story of humanity relies on this goal, to understand nature. Nowadays, because of various applications of the theory of fuzzy differential equations, many researchers are working on fuzzy partial differential equations. Several researchers emphasized studying the precise/numerical solutions of fuzzy differential equations [5–7].

Various analytical and computational methods can solve fuzzy partial differential equations, see for example [8–12]. Integral transform is a very useful technique for solving PDEs and has extensively been used by researchers to solve differential [13]. Fuzzy Laplace transform was defined by [14] and then further developed and used by several authors to solve fuzzy ordinary and fuzzy partial differential equations, see for example [15–18]. Allahviranloo [19], introduced the conformable Laplace transform, and then developed by several researchers to solve conformable differential equations [20,21].

Recently, Younus et al. [22] generalized two predefined concepts under the name fuzzy conformable differential equations, and got the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order Ψ, they used two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order Ψ by a variation of the constant formula.

In this article, we introduce the double fuzzy Laplace transform in the conformable setting, which is more general than the single fuzzy Laplace transform and we extensively used it in the qualitative theory of fuzzy partial differential equations.

1.2 Research Question

In this paper, we discussed the following questions:

1.    In [23] Debnath, provided the solutions of PDEs and Integral and functional equations with double Laplace transform, and Özkan et al. [24], generalized double Laplace transform in the conformable setting. What is the conformable double Laplace transform in the fuzzy environment?

2.    What are the forms of fuzzy partial differential equations (both in 1D and 2D) in conformable cases?

3.    What are the effects of fuzzy conformable Laplace transformation on the solutions of fuzzy conformable PDEs?

4.    What is the application of fuzzy conformable double Laplace transform? Is this transformation providing better results for this application?

1.3 Objective of the Work

A very broad literature including books and papers on the single Laplace transform, its features, and applications are available. However, very few results are available on the double Laplace transform. We generalized the notions of the double Laplace transform in the fuzzy conformable sense. We obtained some basic properties of fuzzy conformable double Laplace. To solve fuzzy conformable partial differential equations, we adopt the fuzzy conformable double Laplace transform.

1.4 Structure of the Study

The organization of this paper is as follows: We present basic principles in Section 2 to use in the main part of the paper. In Section 3, we define the fuzzy double Laplace transform (FDLT), and fuzzy conformable double Laplace transform (FCDLT). Some basic properties of FDLT and FCDLT are also part of Section 3. In Section 4, solutions of the fuzzy conformable partial differential equations are obtained with FCDLT. Concluding remarks are given in Section 5.

2  Basic Concepts

In this section, we recall the basic concepts which we have to use in the major part of the article [14].

A fuzzy set is a map η:R→[0,1] which generalizes classical sets from {0,1} to [0,1]. A fuzzy number η is a fuzzy set that satisfies some additional properties of convexity, normality, upper-semicontinuity, and compact support. We use RΦ to denote the space of all real fuzzy numbers [25]. For 0≤γ<1, γ-cuts for a fuzzy number η is defined as (η,γ)={v∈R:η(v)≥γ}. In γ-cuts form, the fuzzy number η is represented in the form (η,γ)=[(η∗,γ),(η∗,γ)]. A triangular fuzzy number η, denoted by an ordered triple (a,b,c), with the condition a≤b≤c. The γ-cuts associated with triangular fuzzy number η are [a+(b−a)γ,c−(c−b)γ].

If η,υ∈RΦ, then addition on the space of fuzzy numbers by γ-cuts is defined as [(η+υ),γ]=[(η∗,γ)+(υ∗,γ),(η∗,γ)+(υ∗,γ)]. The H-difference for two fuzzy numbers η and υ denoted by η⊖υ and defined as a fuzzy number ω such that ω=η+υ. In γ-cuts form, H-difference for two fuzzy numbers η and υ has the form [(η⊖υ),γ]=[(η∗,γ)−(υ∗,γ),(η∗,γ)−(υ∗,γ)]. A fuzzy-valued function with two variables v and τ assigns an ordered pair (v,τ) to a fuzzy number Φ(v,τ). In γ-cuts form, Φ(v,τ) is represented in the form Φ(v,τ.γ)=[Φ∗(v,τ,γ),Φ∗(v,τ,γ)] ([10]).

A fuzzy-valued function Φ(v,τ) is continuous at any point (v0,τ0) if ‖(v,τ)−(v0,τ0)‖<δ, then we have Φ(v,τ)−L<ϵ. Mathematically, we can write as lim(v,τ)→(v0,τ0)⁡Φ(v,τ)=L.

Before defining fuzzy double Laplace transform, we state the fuzzy single Laplace transform and some relevant properties for the fuzzy-valued function of two variables.

Fuzzy single Laplace transform for Φ(v,τ) with respect to v is defined as

ℓv[Φ(v,τ)]=ϕ(r1,τ)=∫0∞e−r1v⊙Φ(v,τ)dv.

Fuzzy single Laplace transform for Φ(v,τ) with respect to τ is defined as [20]

ℓτ[Φ(v,τ)]=ϕ(v,r2)=∫0∞e−r2τ⊙Φ(v,τ)dτ.

When fuzzy Laplace transform with respect to τ is applied to a strongly generalized partial derivative with respect to v, then we have the result

ℓτ[∂Φ(v,τ)∂v]=∂∂v[(ϕ(v,r2))].

Let us state the translation theorems for fuzzy Laplace transformation:

Theorem 2.1. [14] (First translation theorem.) If Φ is fuzzy Laplace transformable, then

ℓτ(e−aτΦ(v,τ))=ϕ(v,r2+a).

Theorem 2.2. [14] (Second translation theorem.) If Φ is fuzzy Laplace transformable, then

ℓτ[U(v,τ−α)⊙Φ(v,τ−α)]=e−αr2⊙ϕ(v,r2),

where U is the Heaviside function.

Theorem 2.3. For a fuzzy-valued function Φ(v,τ), we have

(∫0∞e−r1v⊙∂Φ∂v(v,τ)dv)=r1⊙ϕ(r1,τ)⊖Φ(0,τ).

Proof. The proof can easily be done using the integration of parts for fuzzy valued function [10].

The following table shows the conformable double Laplace transform for certain functions:

2.1 Strongly Generalized Conformable Partial Derivatives

In this subsection, we define strongly generalized conformable partial derivative to solve fuzzy conformable partial differential equations.

Definition 2.1. For a fuzzy-valued function Φ(v,τ), the strongly generalized conformable partial derivative with respect to v is of order Ψ is defined as a fuzzy number ∂ΨΦ(v,τ)∂vΨ such that

1.    (∀)θ>0, H-differences Φ(v0+θv1−Ψ,τ)⊖Φ(v0,τ) and Φ(v0,τ)⊖Φ(v0−θv1−Ψ,τ) exist and we have

limθ→0⁡Φ(v0+θv1−Ψ,τ)⊖Φ(v0,τ)θ=limθ→0⁡Φ(v0,τ)⊖Φ(v0−θv1−Ψ,τ)θ.

2.    (∀)θ>0, there exist H-differences Φ(v0,τ)⊖Φ(v0+θv1−Ψ,τ) and Φ(v0−θv1−Ψ,τ)⊖Φ(v0,τ) and we have

limθ→0⁡Φ(v0,τ)⊖ΦΨ(v0+θv1−Ψ,τ)−θ=limθ→0⁡Φ(v0−θv1−Ψ,τ)⊖Φ(v0,τ)−θ.

Proposition 2.1. The fuzzy-valued function Φ(v,τ) is said to be differential of type (Ψ−1) if Φ is differentiable in the first form of the above definition, and differential of type (Ψ−2) if Φ is differentiable in the second form.

Definition 2.2. For a fuzzy-valued function Φ(v,τ), the strongly generalized conformable partial derivative of order δ with respect to τ is defined as a fuzzy number ∂δΦ(v,τ)∂τδ such that

1.    (∀)θ>0, H-differences Φ(v,τ0+θτ1−δ)⊖Φ(v,τ0) and Φ(v,τ0)⊖Φ(v,τ0−θτ1−δ) exist, and we have

limθ→0⁡Φ(v,τ0+θτ1−δ)⊖Φ(v,τ0)θ=limθ→0⁡Φ(v,τ0)⊖Φ(v,τ0−θτ1−δ)θ.

2.    (∀)θ>0, H-differences Φ(v,τ0)⊖Φ(v,τ0+θτ1−δ) and Φ(v,τ0−θτ1−δ)⊖Φ(v,τ0) exist, and we have

limθ→0⁡Φ(v,τ0)⊖Φ(v,τ0+θτ1−δ)−θ=limθ→0⁡Φ(v,τ0−θτ1−δ)⊖Φ(v,τ0)−θ.

Proposition 2.2. Φ is said to be differential of type (δ−1) if Φ is differentiable in the first form, and differential of type (δ−2) if Φ is differentiable in the second form.

For a fuzzy-valued function Φ, the fuzzy conformable integral of order Ψ is defined as

IΨΦ(v)=∫0vΦ(μ)μΨ−1dμ,

where integration is in the sense of fuzzy Riemann integral.

Lemma 2.1. If a continuous fuzzy-valued function Φ(v,τ) is strongly generalized conformable partial differentiable with respect to τ, we have

∫ab∂δΦ(v,τ)∂τδτδ−1dτ=Φ(v,b)⊖Φ(v,a).

Lemma 2.2. For a continuous fuzzy-valued function Φ(v,τ) which is strongly generalized conformable partial differentiable, we have

∫ab∂ΨΦ(v,τ)∂vΨvΨ−1dv=Φ(b,τ)⊖Φ(a,τ).

Proof. If Φ(v,τ) is differentiable of type (Ψ−1), then we have

∫ab∂ΨΦ(v,τ,γ)∂vΨvΨ−1dv=∫ab[∂ΨΦ∗(v,τ,γ)∂vΨ,∂ΨΦ∗(v,τ,γ)∂vΨ]vΨ−1dv,=[Φ∗(b,τ,γ)−Φ∗(a,τ,γ),Φ∗(b,τ,γ)−Φ∗(a,τ,γ)],=Φ(b,τ)⊖Φ(a,τ).

This completes the proof.

3  Double Laplace Transform

Now, we move towards our main results on double Laplace transform.

3.1 Fuzzy Double Laplace Transform

For this, we first define fuzzy double Laplace transform and some related properties.

Fuzzy double Laplace transform of a fuzzy-valued function Φ(v,τ) is

ℓvℓτ[Φ(v,τ)]=ϕ(r1,r2)=∫0∞∫0∞e−r2τ⊙e−r1v⊙Φ(v,τ)dvdτ,(1)

where the integral in the definition should converge.

We can write the above definition in the form

ℓvℓτ[Φ(v,τ)]=[ℓvℓτ[Φ∗(v,τ,γ)],ℓvℓτ[Φ∗(v,τ,γ)]].

Definition 3.1. Fuzzy double inverse Laplace transform is defined as

ℓv−1ℓτ−1[ϕ(r1,r2)]=Φ(v,τ)=14π2∫α−ι∞α+ι∞∫β−ι∞β+ι∞er2τer1v⊙ϕ(r1,r2)dr1dr2.

Fuzzy double Laplace transform is linear, i.e., If ℓvℓτ[Φ(v,τ)]=ϕ(r1,r2), then for any constants α,β and fuzzy-valued functions Φ and ψ, we have

ℓvℓτ[α⊙Φ(v,τ)+β⊙ψ(v,τ)]=α⊙ℓvℓτ[Φ(v,τ)]+β⊙ℓvℓτ[ψ(v,τ)].

Similarly, the fuzzy inverse double Laplace transform is also linear.

While studying the theory of fuzzy Laplace transform, we have to study the absolute value of the fuzzy-valued function.

Definition 3.2. For the fuzzy-valued funcion Φ, the absolute value of the fuzzy-valued function in the γ-cuts form as

[Φ(v,τ,γ)]=[|Φ∗(v,τ,γ)|,|Φ∗(v,τ,γ)|].

Definition 3.3. A fuzzy-valued function Φ is called of exponential order in the fuzzy sense if

Φ(v,τ)≤Meαv+βτ,(∀)α,β,M∈R+.

Remark 3.1. Fuzzy double Laplace transform does not exist for all fuzzy-valued functions. For example, Φ(v,τ)=vτ⊙η or v2+τ2⊙η is not the fuzzy double Laplace transform of any fuzzy-valued function Φ(v,τ) because Φ(v,τ) does not converge to zero whenever v→∞,τ→∞. Also, fuzzy double Laplace transform for Φ(v,τ)=exp⁡(αv2+βτ2)⊙η with α,β>0 does not exist since it is not of exponential order because

limv→∞,τ→∞⁡exp⁡(αv2+βτ2−r1v2−r2τ2)⊙η=∞.

Here we give the condition for the existence of fuzzy double Laplace transform.

Theorem 3.1. If a fuzzy-valued function Φ satisfies two conditions:

1.    Φ is of fuzzy exponential order.

2.    Φ is bounded and piecewise continuous, then fuzzy double Laplace transform exists and alsoconverges absolutely.

Proof. Given is Φ is bounded, so we have |Φ(v,τ)|≤M1. Also, Φ has exponential order, so by definition of exponential order in the fuzzy sense, we have

|Φ(v,τ)|≤M2eαv+βτ,(∀)M2,α,β∈R+.

Put M=max{M1,M2}, we obtain

|Φ(v,τ)|≤Meαv+βτ,(∀)α,β,M∈R+.

This yields

∫0∞∫0∞e−r1v−r2τ⊙|Φ(v,τ)|dvdτ≤M∫0∞∫0∞e−v(r1−α)e−τ(r2−β)dvdτ.

Thus we have

limr1→∞,.r2→∞⁡ϕ(r1,r2)=M(r1−α)(r2−β),forr1>α,r2>β.

Theorem 3.2. Fuzzy double Laplace transform for a fuzzy-valued function Φ differentiable of the first-order is

Case 1: When Φ is strongly generalized partial differentiable with respect to v, we have

1.     If Φ is differentiable of type (1), then

ℓvℓτ[∂Φ∂v(v,τ)]=r1⊙ϕ(r1,r2)⊖ϕ(0,r2).

2.     If Φ is differentiable of type (2), then

ℓvℓτ[∂Φ∂v(v,τ)]=⊖[ϕ(0,r2)−r1⊙ϕ(r1,r2)].

Case 2: When Φ is strongly generalized partial differentiable with respect to τ, we have

1.     If Φ is differentiable of type (1), then

2.     ℓvℓτ[∂Φ∂τ(v,τ)]=r2⊙ϕ(r1,r2)⊖ϕ(r1,0).

3.     If Φ is differentiable of type (2), then

ℓvℓτ[∂Φ∂τ(v,τ)]=⊖[ϕ(r1,0)−r1⊙ϕ(r1,r2)].

Proof. By using the definition of fuzzy double Laplace transform, we have

ℓvℓτ[∂Φ∂v(v,τ)]=∫0∞e−r2τ⊙(∫0∞e−r1v⊙∂Φ∂v(v,τ)dv)dτ.(2)

Using Theorem 2.3, we have

(∫0∞e−r1v⊙∂Φ∂v(v,τ)dv)=r1⊙ϕ(r1,τ)⊖Φ(0,τ).(3)

Using Eq. (3) in the Eq. (2), we have

ℓvℓτ[∂Φ∂v(v,τ)]=r1⊙ϕ(r1,r2)⊖ϕ(0,r2).

This completes our proof.

Theorem 3.3. For second-order fuzzy partial derivative with respect to v, fuzzy double Laplace transform is

Case 1: When Φ is differentiable with respect to v, we have

1.    If Φ and ∂Φ(v,τ)∂v both are differentiable of type (1), we have

ℓvℓτ[∂2Φ∂v2(v,τ)]=r12⊙ϕ(r1,r2)⊖r1⊙ϕ(0,r2)⊖∂ϕ(0,r2)∂v.

2.    If Φ is differentiable of type (1) and ∂Φ(v,τ)∂v is differentiable of type (2), then

ℓvℓτ[∂2Φ∂v2(v,τ)]=−∂ϕ(0,r2)∂v⊖(−r12⊙ϕ(r1,r2))⊖r1⊙ϕ(0,r2).

3.    If Φ is differentiable of type (2) and ∂Φ(v,τ)∂v is differentiable of type (1), then

ℓvℓτ[∂2Φ∂v2(v,τ)]=−r1⊙ϕ(0,r2)⊖(−r12⊙ϕ(r1,r2))⊖∂ϕ(0,r2)∂v.

4.    If both Φ and ∂Φ(v,τ)∂v are differentiable of type (2), we have

ℓvℓτ[∂2Φ∂v2(v,τ)]=r12⊙ϕ(r1,r2)⊖r1⊙ϕ(0,r2)−∂ϕ(0,r2)∂v.

Case 2: For second-order partial derivative with respect to τ, double Laplace transform is

1.    If Φ and ∂Φ∂τ(v,τ) both are differentiable of type (1), we have

ℓvℓτ[∂2Φ∂τ2(v,τ)]=r22⊙ϕ(r1,r2)⊖r2⊙ϕ(r1,0)⊖∂ϕ(r1,0)∂τ.

2.    If Φ is differentiable of type (1) and ∂Φ∂τ(v,τ) is differentiable of type (2), then

ℓvℓτ[∂2Φ∂τ2(v,τ)]=−∂ϕ(r1,0)∂τ⊖(−r22⊙ϕ(r1,r2))⊖r2⊙ϕ(r1,0).

3.    If Φ is differentiable of type (2) and ∂Φ∂τ(v,τ) is differentiable of type (1), then

ℓvℓτ[∂2Φ∂τ2(v,τ)]=−r2⊙ϕ(r1,0)⊖(−r22⊙ϕ(r1,r2))⊖∂ϕ(r1,0)∂τ.

4.    If Φ and ∂Φ∂τ(v,τ) both are differentiable of type (2), we have

ℓvℓτ[∂2Φ∂τ2(v,τ)]=r22⊙ϕ(r1,r2)⊖r2⊙ϕ(r1,0)−∂ϕ(r1,0)∂τ.

Theorem 3.4. For a fuzzy-valued function differentiable with respect to v and τ, fuzzy double Laplace transform is

1.    When Φ(v,τ) is differentiable of type (1) with respect to v and τ, we have

ℓvℓτ[∂2Φ∂v∂τ(v,τ)]=r1r2⊙ϕ(r1,r2)⊖r1⊙ϕ(r1,0)⊖r2⊙ϕ(0,r2)+Φ(0,0).

2.    When Φ(v,τ) is differentiable of type (2) with respect to v and τ, we have

ℓvℓτ[∂2Φ∂v∂τ(v,τ)]=Φ(0,0)⊖r2⊙ϕ(0,r2)−⊖[(−r2⊙ϕ(r1,0)−⊖r1r2⊙ϕ(r1,r2))].

3.    When Φ(v,τ) is differentiable of type (1) with respect to v and differentiable of type (2) with respect to τ, we have

ℓvℓτ[∂2Φ∂v∂τ(v,τ)]=[−⊖r1r2⊙ϕ(r1,r2)−r1⊙ϕ(r1,0)]⊖−r2⊙ϕ(0,r2)−⊖Φ(0,0).

4.    When Φ(v,τ) is differentiable of type (2) with respect to v and differentiable of type (1) with respect to τ, we have

ℓvℓτ[∂2Φ∂v∂τ(v,τ)]=−⊖[r1r2⊙ϕ(r1,r2)⊖r1⊙ϕ(r1,0)]−[r2⊙ϕ(0,r2)−⊖Φ(0,0)].

Proof. We provide proof for case (2) here. Other cases are similar.

Since Φ is differentiable of type (2) with respect to v, then

ℓv[∂Φ∂v(v,τ)]=−Φ(0,τ)−⊖r1⊙ϕ(r1,τ).(4)

If ∂Φ∂v is differentiable of type (2) with respect to τ, then

ℓvℓτ[∂2Φ∂v∂τ(v,τ)]=ℓτ[∂∂τ(−Φ(0,τ))−⊖∂∂τ(r1⊙ϕ(r1,τ))].(5)

Now, apply the Laplace transform for τ, we have

ℓvℓτ[∂2Φ∂v∂τ(v,τ)]=Φ(0,0)⊖r2⊙ϕ(0,r2)−⊖[(−r2⊙ϕ(r1,0)−⊖r1r2⊙ϕ(r1,r2))].

This completes the proof.

Theorem 3.5. For a fuzzy-valued function, whose fuzzy double Laplace transform exists, we have

ℓvℓτ[eαv+βτ⊙Φ(v,τ)]=ϕ(r1−α,r2−β).

Proof. Using the definition of fuzzy double Laplace transform, we have

ℓvℓτ[eαv+βτ⊙Φ(v,τ)]=∫0∞∫0∞e−r2τe−r1veαv+βτ⊙Φ(v,τ)dvdτ.

This results in

ℓvℓτ[eαv+βτ⊙Φ(v,τ)]=∫0∞∫0∞e−(r2−β)τe−(r1−α)v⊙Φ(v,τ)dvdτ.

Thus, we obtain our required result.

Theorem 3.6. For a fuzzy-valued function, whose fuzzy double Laplace transform exists, we have

ℓvℓτ[vτ⊙Φ(v,τ)]=(−1)1+1⊙∂2ϕ(r1,r2)∂r1∂r2.

Proof. If we take the strongly generalized partial derivative with respect to r1 and r2, we have

∂2ϕ(r1,r2)∂r1∂r2=(−1)(−1)∫0∞∫0∞e−r2τe−r1v⊙Φ(v,τ)dvdτ.

This can be written as

∂2ϕ(r1,r2)∂r1∂r2=(−1)1+1⊙ℓvℓτ[vτΦ(v,τ)],

or we can write

ℓvℓτ[vτΦ(v,τ)]=(−1)1+1⊙∂2ϕ(r1,r2)∂r1∂r2.

In general, we have

ℓvℓτ[vpτqΦ(v,τ)]=(−1)p+q⊙∂p+qϕ(r1,r2)∂r1p∂r2q.

Theorem 3.7. (Second translation theorem). For a fuzzy-valued function, whose fuzzy double Laplace transform exists, the second translation theorem in the fuzzy sense has the form

ℓvℓτ[Φ(v−η,τ−μ)⊙U(v−η,τ−μ)]=e−r1η−r2μ⊙ϕ(r1,r2),

where U(v,τ) is the Heaviside unit step function defined by

U(v−η,τ−μ)=1,whenv>η,andτ>μ,U(v−η,τ−μ)=0,whenv<ηandτ<μ.

Proof. Using the definition of fuzzy double Laplace transform, we have

ℓvℓτ[Φ(v−η,τ−μ)⊙U(v−η,τ−μ)]=∫0∞∫0∞e−r1v−r2τ⊙Φ(v−η,τ−μ)⊙U(v−η,τ−μ)dvdτ,=∫η∞∫μ∞e−r1v−r2τ⊙Φ(v−η,τ−μ)dvdτ.

Put v−η=a,τ−μ=b, we get

ℓvℓτ[Φ(v−η,τ−μ)⊙U(v−η,τ−μ)]=e−r1η−r2μ⊙∫0∞∫0∞e−r1a−r2b⊙Φ(a,b)dadb,=e−r1η−r2μ⊙ϕ(r1,r2).

Definition 3.4. If Φ(v,τ) is a fuzzy-valued function and ψ(v,τ) is a real-valued function, then convolution in fuzzy sense is defined as

(Φ∘∘ψ)(v,τ)=∫0v∫0τΦ(q,r)⊙ψ(v−q,τ−r)dqdr.

Theorem 3.8. For fuzzy double Laplace transform, the Convolution theorem is given by

ℓvℓτ[(Φ∘∘ψ)(v,τ)]=ℓvℓτ[Φ(v,τ)]⊙ℓvℓτ[ψ(v,τ)].

3.2 Fuzzy Conformable Double Laplace Transform

Now, we generalized the concept of fuzzy double Laplace transform to fuzzy conformable double Laplace transform.

Definition 3.5. Fuzzy conformable double Laplace transform for a fuzzy-valued function Φ(v,τ) is

ℓΨvℓδτ[Φ(v,τ)]=ϕ(r1,r2)=∫0∞∫0∞e−r2τδδe−r1vΨΨ⊙Φ(v,τ)vΨ−1τδ−1dvdτ,

where the integral in the definition should converge.

We can write the above definition in the form

ℓΨvℓδτ[Φ(v,τ)]=[ℓΨvℓδτ[Φ∗(v,τ)],ℓΨvℓδτ[Φ∗(v,τ)]].

Fuzzy conformable double Laplace transform is linear. i.e., for constants α and β and fuzzy-valued functions Φ(v,τ) and ψ(v,τ), we have

ℓΨvℓδτ[α⊙Φ(v,τ)+β⊙ψ(v,τ)]=α⊙ℓΨvℓδτ[Φ(v,τ)]+β⊙ℓΨvℓδτ[ψ(v,τ)].

Definition 3.6. Fuzzy conformable inverse double Laplace transform is defined as

ℓΨv−1ℓδτ−1[ϕ(r1,r2)]=Φ(v,τ)=14π2∫α−ι∞α+ι∞∫β−ι∞β+ι∞er1vΨΨ⊙er2τδδ⊙ϕ(r1,r2)vΨ−1τδ−1dr1dr2.

Although the fuzzy conformable double Laplace transform exists for a large variety of fuzzy-valued functions, it does not always exist. For example, fuzzy conformable double Laplace for Φ(v,τ)=η⊙ev2Ψ2Ψ+τ2δ2δ does not exist because the integral does not converge.

Here we give the criteria for the existence of fuzzy conformable double Laplace transform.

Definition 3.7. A fuzzy-valued function Φ(v,τ) is of exponential order, if for some real constants α,β, we obtain supv,τ>0⁡|Φ(v,τ)|≤MeαvΨΨ+βτδδ.

Theorem 3.9. Let Φ(v,τ) be of exponential order and continuous on the interval [0,∞), then the fuzzy conformable double Laplace transform of Φ exists.

Proof. Since Φ is of exponential order, so we have

|Φ(v,τ)|≤MeαvΨΨ+βτδδ,

Now, we have

∫0∞∫0∞e−r2τδδe−r1vΨΨ⊙|Φ(v,τ)|vΨ−1τδ−1dvdτ≤M∫0∞∫0∞e−vΨΨ(r1−α)e−τδδ(r2−β)vΨ−1τδ−1dvdτ.

Now, after performing fuzzy conformable integration and taking limr1→∞,.r2→∞, we have

∫0∞∫0∞e−r1vΨΨe−r2τδδ⊙|Φ(v,τ)|vΨ−1τδ−1dvdτ≤M(r1−α)(r2−β),forr1>α,r2>β.

So we have

limr1→∞,.r2→∞⁡ϕ(r1,r2)=0.

Thus proved.

Relation between fuzzy double Laplace transform and fuzzy conformable double Laplace transform is

Lemma 3.1. ℓΨvℓδτ[Φ(v,τ)]=ℓvℓτ[ϕ((Ψv)1Ψ,(δτ)1δ)(r1,r2)].

Proof. From Definition 3.5, we have

ℓΨvℓδτ[Φ(v,τ)]=∫0∞∫0∞e−r2τδδe−r1vΨΨ⊙Φ(v,τ)vΨ−1τδ−1dvdτ.

Substitute τδδ=t,vΨΨ=u, we have

ℓΨvℓδτ[Φ(v,τ)]=∫0∞∫0∞e−r2te−r1u⊙Φ((uΨ)1Ψ,(tδ)1δ)dtdu,=ℓvℓτ[ϕ((Ψv)1Ψ,(δτ)1δ)(r1,r2)].