# On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations

Authors: ElHassan ElJaoui; Said Melliani
Journal: International Journal of Differential Equations
 (2016)

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## Abstract

We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.


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## Body

## 1. Introduction

Wu introduced in [1] the improper fuzzy Riemann integral and presented some of its elementary properties; then he studied numerically this kind of integrals.This notion was exploited by certain researchers to study fuzzy differential equations (FDEs) of first or second order utilizing fuzzy Laplace transform, namely, by Allahviranloo and Ahmadi in [2], then by Salahshour et al. (see [3, 4]), and by ElJaoui et al. in [5].The objective of this paper is to study the improper fuzzy Riemann integrals by establishing some important results about the continuity and the differentiability of a fuzzy improper integral depending on a given parameter.These results are then employed to prove some fuzzy Laplace transform’s properties, which we use to solve fuzzy partial differential equations (FPDEs).The organization of the remainder of this work is as follows. Section2 is reserved for preliminaries. In Section 3, the main results are proved and new properties of fuzzy Laplace transform are investigated. Then, in Section 4, the procedure for solving first-order FPDEs by fuzzy Laplace transform is proposed. Section 5 deals with some numerical examples. In Section 6, we present conclusion and a further research topic.


## 2. Preliminaries

ByPc(R) we meant the set of all nonempty compact convex subsets of R, which is endowed with the usual addition and scalar multiplication. Denote (see [6]) (1)E=μ:R⟶0,1∣μverifies1–4below,where(1)
μis normal; that is, ∃t∈R for which μ(t)=1,(2)
μ is convex in the fuzzy sense,(3)
μ is upper semicontinuous,(4)
the closure of its supportsupp⁡μ=t∈R∣μ(t)>0 is compact.For 0<α≤1, μα=t∈R∣μ(t)≥α denotes the α-level set of μ∈E.Then, it is obvious thatμα∈Pc(R) for all μ∈E, 0≤α≤1, and(2)μ1+μ2α=μ1α+μ2α,kμα=kμα.Let D:E×E→[0,∞) be a function which is defined by the identity (3)Dμ1,μ2=sup0≤α≤1⁡dμ1α,μ2α,where d is the Hausdorff distance defined in Pc(R). Then, it is clear that E,D is a complete metric space (for more details about the metric D see [7]).Definition 1 (see [2]).
One defines a fuzzy numberv in parametric form as a couple (v_,v¯) of mappings v_(α) and v¯(α), 0≤α≤1, verifying the following properties: (1)
v_(α) is bounded increasing left continuous in ]0,1] and right continuous at 0.(2)
v¯(α) is bounded decreasing left continuous in ]0,1] and right continuous at 0.(3)
v_(α)≤v¯(α) for all 0≤α≤1.The following identity holds true (see [8]):(4)Dμ1,μ2=sup0≤α≤1⁡max⁡μ_1α-μ_2α,μ¯1α-μ¯2α.Theorem 2 (see [1]).
One considers a fuzzy valued functionF(x)=(F_(x,α),F¯(x,α)) defined on [a,∞[. Suppose that, for all fixed α∈[0,1], the crisp functions F_(x,α),F¯(x,α) are integrable on [a,b], for every b≥a, and that there exist two positive constants K_(α) and K¯(α) such that ∫ab‍|F_(x,α)|dx≤K_(α) and ∫ab‍|F¯(x,α)|dx≤K¯(α) for every b≥a. Then F(x) is fuzzy Riemann integrable (in the sense of Wu) on [a,∞[, its improper fuzzy integral ∫a∞‍F(x)dx∈E, and(5)∫a∞Fxdx=∫a∞F_x,αdx,∫a∞F¯x,αdx.For μ1,μ2∈E, if there exists an element μ3∈E such that μ3=μ1+μ2, then μ3 is called the Hukuhara difference of μ1 and μ2, which we denote by μ1⊖μ2.Definition 3 (see [2]).
A mappingF:(a,b)→E is said to be strongly generalized differentiable at x∈(a,b), if there exists F′(x)∈E, such that (i)
for allh>0  being very small, there exist F(x+h)⊖F(x); F(x)⊖F(x-h); and the limits (6)limh→0+⁡Fx+h⊖Fxh=limh→0+⁡Fx⊖Fx-hh=F′x,or(ii)
for allh>0  being very small, there exist F(x)⊖F(x+h); F(x-h)⊖F(x); and the limits (7)limh→0+Fx⊖Fx+h-h=limh→0+Fx-h⊖Fx-h=F′xor(iii)
for allh>0  being very small, there exist F(x+h)⊖F(x); F(x-h)⊖F(x); and the limits (8)limh→0+Fx+h⊖Fxh=limh→0+Fx-h⊖Fx-h=F′x,or(iv)
for allh>0  being very small, there exist F(x)⊖F(x+h); F(x)⊖F(x-h); and the limits (9)limh→0+Fx⊖Fx+h-h=limh→0+Fx⊖Fx-hh=F′x.The next theorem permits us to consider only case (i) or case (ii) of Definition3 almost everywhere in the domain of the mappings studied.Theorem 4 (see [9]).
IfF:(a,b)→E is a strongly generalized differentiable function on (a,b) in the sense of Definition 3, (iii) or (iv), then F′(x)∈R for each x∈(a,b).Theorem 5 (see, e.g., [10]).
We consider a fuzzy functionF:R→E which is represented by F(x)=(F_(x,α),F¯(x,α)), for all α∈[0,1]: (1)
IfF is (i)-differentiable, then the crisp functions F_(x,α) and F¯(x,α) are differentiable and F′(x)=(F_′(x,α),F¯′(x,α)).(2)
IfF is (ii)-differentiable, then the crisp functions F_(x,α) and F¯(x,α) are differentiable and F′(x)=(F¯′(x,α),F_′(x,α)).Definition 6 (see [2]).
IfF:[0,∞[→E is a continuous mapping such that e-sxF(x) is fuzzy Riemann integrable on [0,∞[ then ∫0∞‍e-sxF(x)dx is called the fuzzy Laplace transform of F which one denotes by (10)LFx=∫0∞e-sxFxdx,s>0.
Denote byL(k(x)) the classical Laplace transform of a crisp function k(x), and then (11)LFx=LF_x,α,LF¯x,α.Theorem 7 (see [2]).
LetF:[0,∞[→E be a fuzzy valued function and F′ its derivative on [0,∞[. Then, if F is (i)-differentiable(12)LF′x=sLFx⊖F0or if F is (ii)-differentiable(13)LF′x=-F0⊖-sLFxprovided that the Laplace transforms of F and F′ exist.


## 3. Continuity and Differentiability of Fuzzy Improper Integral

In this section,I denotes one of the intervals, -∞,b or b,∞ or -∞,∞, where b∈R, J denotes another interval, and A is a nonempty subset of R.Lemma 8.
Letf1(t), f2(t) be two fuzzy valued functions, which are fuzzy Riemann integrable on I, in the sense of Wu (see [1]), such that the real function D(f1(t),f2(t)) is integrable on I, and then (14)D∫If1tdt,∫If2tdt≤∫IDf1t,f2tdt.Proof.
From identity (4), we have(15)D∫If1tdt,∫If2tdt=sup0≤α≤1⁡max⁡∫If_1t,α-f_2t,αdt,∫If¯1t,αdt-f¯2t,αdt≤sup0≤α≤1⁡max⁡∫If_1t,α-f_2t,αdt,∫If¯1t,α-f¯2t,αdt≤sup0≤α≤1⁡∫Imax⁡f_1t,α-f_2t,α,f¯1t,α-f¯2t,αdt≤∫Isup0≤α≤1⁡max⁡f_1t,α-f_2t,α,f¯1t,α-f¯2t,αdt=∫IDf1t,f2tdt.Theorem 9.
LetF(x,t):A×I→E be a fuzzy function, satisfying the following conditions: (H1)
For allx∈A, t↦F(x,t) is continuous on I.(H2)
For eacht∈I, x↦F(x,t) is continuous on A⊂R.(H3)
For allα∈[0,1] there exist a couple of nonnegative, continuous crisp functions φα(t) and ψα(t), which are integrable on I verifying, for all x∈A, t∈I:(16)F_x,t,α≤φαt,F¯x,t,α≤ψαt.Therefore, the fuzzy mapping ϕ(x)=∫I‍F(x,t)dt is continuous on A.Proof.
Letx∈A and let {xk}k=1∞ be a sequence of elements of A, which converges to x as k→∞. For k∈N, t∈I, and α∈[0,1], we have(17)F_xk,t,0≤F_xk,t,α≤F_xk,t,1,F¯xk,t,1≤F¯xk,t,α≤F¯xk,t,0.Thus(18)F_xk,t,α≤max⁡F_xk,t,1,F_xk,t,0≤max⁡φ0t,φ1t=gt,F¯xk,t,α≤max⁡F¯xk,t,1,F¯xk,t,0≤max⁡ψ0t,ψ1t=ht.By tending k→∞ and using assumption (H2), we obtain(19)F_x,t,α≤max⁡φ0t,φ1t=gt,F¯x,t,α≤max⁡ψ0t,ψ1t=ht.Therefore(20)DFxk,t,Fx,t=sup0≤α≤1⁡max⁡F_xk,t,α-F_x,t,α,F¯xk,t,α-F¯x,t,α,DFxk,t,Fx,t≤2gt+ht.From (H1) and (H3), we deduce that the mappings g(t), h(t), and DF(xk,t),F(x,t) are all integrable on I.
On the other hand, we get the following inequality from Lemma8:(21)D∫IFxk,tdx,∫IFx,tdx≤∫IDFxk,t,Fx,tdx.That is,(22)Dϕxk,ϕx≤∫IDFxk,t,Fx,tdx.By assumption (H2), we have D(F(xk,t),F(x,t))→0 as k→∞.
So, by the dominated convergence theorem,∫I‍D(F(xk,t),F(x,t))dx→0 as k→∞.
From inequality (22), we deduce that ϕ(xk)→ϕ(x) as k→∞.
Consequently,ϕ is continuous on A.Lemma 10.
One considers two fuzzy valued functionsf1(t), f2(t):I→E, which are fuzzy Riemann integrable on I (in the sense of Wu), such that f1(t)⊖f2(t) exists for all t∈I, then f1(t)⊖f2(t) is fuzzy Riemann integrable on I, the Hukuhara difference ∫I‍f1(t)dt⊖∫I‍f2(t)dt is well defined, and (23)∫If1t⊖f2tdx=∫If1tdt⊖∫If2tdt.Proof.
Letk(t)=f1(t)⊖f2(t); that is, f1(t)=f2(t)+k(t). It is clear that there exist positive constants K_(α,f1), K¯(α,f1), K_(α,f2), and K¯(α,f2) such that, for all a≤b in I, we have(24)∫abf_1t,αdt≤K_α,f1,∫abf¯1t,αdt≤K¯α,f1,∫abf_2t,αdt≤K_α,f2,∫abf¯2t,αdt≤K¯α,f2.Hence(25)∫abk_t,αdt=∫abf_1t,α-f_2t,αdt≤K_α,f1+K_α,f2and similarly(26)∫abk¯t,αdt=∫abf¯1t,α-f¯2t,αdt≤K¯α,f1+K¯α,f2.Then from Theorem 2, k(t) is fuzzy Riemann integrable on I.
By “linearity” of the fuzzy integral, we get(27)∫If1tdt=∫If2tdt+∫Iktdt.Thus, ∫I‍f1(t)dt⊖∫I‍f2(t)dt exists and ∫I‍f1(t)dt⊖∫I‍f2(t)dt=∫I‍k(t)dt.Theorem 11.
One considers a fuzzy valued functionF(x,t):J×I→E, verifying the following assumptions: (A1)
For allx∈J, t↦F(x,t) is continuous and fuzzy Riemann integrable on I.(A2)
For allt∈I, x↦F(x,t) is (i)-differentiable on the interval J.(A3)
For allx∈J, t↦∂F/∂x(x,t) is continuous on I.(A4)
For allt∈I, x↦∂F/∂x(x,t) is continuous on J.(A5)
For allα∈[0,1] there exist a couple of continuous crisp functions φα(t) and ψα(t), which are integrable on I verifying, for all x∈J, t∈I: (28)∂F_∂xx,t,α≤φαt,∂F¯∂xx,t,α≤ψαt.Therefore, the fuzzy mapping ϕ(x)=∫I‍F(x,t)dt is (i)-differentiable on J and(29)ϕ′x=∫I∂F∂xx,tdt,∀x∈J.Moreover, if one replaces assumption (A2) by the alternative condition (A2′)
for allt∈I, x↦F(x,t) is (ii)-differentiable on J,then the fuzzy function ϕ(x) is (ii)-differentiable on J and (29) remains true.Proof.
Assume that (A1)–(A5) hold true. Let x∈J, ξ0>0 being very small, and define the auxiliary functions (30)g1ξ,t=Fx+ξ,t⊖Fx,tξ,ξ∈0,ξ0∂F∂xx,t,ξ=0,g2ξ,t=Fx,t⊖Fx-ξ,tξ,ξ∈0,ξ0∂F∂xx,t,ξ=0.For fixed ξ∈0,ξ0, we have(31)ϕx+ξ⊖ϕxξ=1ξ∫IFx+ξ,tdt⊖∫IFx,tdt=∫IFx+ξ,t⊖Fx,tξdt=∫Ig1ξ,tdt,where the existence of the Hukuhara differences is ensured by the (i)-differentiability of x↦F(x,t) and by Lemma 10.
Analogously, we get(32)ϕx⊖ϕx-ξξ=∫Ig2ξ,tdt.From assumptions (A1)–(A4), we deduce that g1 and g2 satisfy conditions (H1)-(H2) of Theorem 9.
On the other hand, using the finite increments theorem, we obtain(33)g_1ξ,t,α=F_x+ξ,t,α-F_x,t,αξ≤sup0≤v≤ξ0⁡∂F_∂xx+v,t,α≤φαt,g¯1ξ,t,α=F¯x+ξ,t,α-F¯x,t,αξ≤sup0≤v≤ξ0⁡∂F¯∂xx+v,t,α≤ψαt.Similarly, we have(34)g_2ξ,t,α=F_x,t,α-F_x-ξ,t,αh≤sup0≤v≤ξ0⁡∂F_∂xx-v,t,α≤φαt,g¯2ξ,t,α=F¯x,t,α-F¯x-ξ,t,αh≤sup0≤v≤ξ0⁡∂F¯∂xx-v,t,α≤ψαt.Inequalities (33) and (34), which are obviously also true for ξ=0, ensure that g1 and g2 satisfy condition (H3) of Theorem 9.
Applying the latter theorem, we get(35)limξ→0+⁡ϕx+ξ⊖ϕxξ=∫Ig10,tdt=∫I∂F∂xx,tdt,limξ→0+⁡ϕx⊖ϕx-ξξ=∫Ig20,tdt=∫I∂F∂xx,tdt.Therefore, ϕ is (i)-differentiable at x and(36)ϕ′x=∫I∂F∂xx,tdt.The proof under assumption (A2′) instead of (A2) is similar to the first case.Theorem 12.
One considers a fuzzy functionu(ξ,τ):0,∞×0,∞→E. Suppose that the mapping F(ξ,τ)=e-sτu(ξ,τ) satisfies assumptions (A1)–(A5) above, for all s≥s0 for some s0>0.
LetLτ[u(ξ,τ)] or L[u(ξ,τ)] (for short) denote the fuzzy Laplace transform of u(ξ,τ) with respect to the time variable τ. Then (37)Lτuξξ,τ=∂∂ξLτuξ,τ.Proof.
For fixeds≥s0, then using Theorem 11 we have(38)Lτuξξ,τ=∫0∞e-sτuξξ,τdτ=∫0∞Fξξ,τdτ=∂∂ξ∫0∞Fξ,τdτ,Lτuξξ,τ=∂∂ξLτuξ,τ.Theorem 13.
Letu(ξ,τ) be a fuzzy valued function on 0,∞×0,∞ into E. Suppose that the mappings τ↦F(ξ,τ)=e-sτu(ξ,τ) and τ↦G(ξ,τ)=e-sτuτ(ξ,τ) are fuzzy Riemann integrable on 0,∞, for all s≥s0 for some s0>0. Consider the following:(a)
Ifu(ξ,τ) is (i)-differentiable with respect to τ, then(39)Lτuτξ,τ=sLτuξ,τ⊖uξ,0,(b)
Ifu(x,τ) is (ii)-differentiable with respect to τ, then(40)Lτuτξ,τ=-uξ,0⊖-sLτuξ,τ.Proof.
This is a direct result of Theorem12, by fixing ξ≥0 and taking the Laplace transforms with respect to τ.


## 4. Fuzzy Laplace Transform Algorithm for First-Order Fuzzy Partial Differential Equations

Our aim now is to solve the following first-order FPDE using the fuzzy Laplace transform method under strongly generalized differentiability:(41)uξξ,τ+auτξ,τ=fξ,τ,uξ,τ,uξ,0=gξ,u0,τ=hτ,where u(ξ,τ) is a fuzzy function of ξ≥0, τ≥0, a is a real constant, and f(ξ,τ,u), g(ξ), and h(τ) are fuzzy valued functions, such that f(ξ,τ,u) is linear with respect to u. For short, assume that a≥0 (case a<0 is similar).By using fuzzy Laplace transform with respect toτ, we get(42)Lτuξξ,τ+aLτuτξ,τ=Lτfξ,τ,uξ,τ.Therefore, we have to distinguish the following cases for solving (42):(a)
Case1: If u is (i)-differentiable with respect to ξ and τ, then by Laplace transform(43)Lu_ξξ,τ,α+aLu_τξ,τ,α=Lf_ξ,τ,uξ,τ,Lu¯ξξ,τ,α+aLu¯τξ,τ,α=Lf¯ξ,τ,uξ,τ,wheref_(ξ,τ,u(ξ,τ),α)=min{f(ξ,τ,v)/v∈(u_(ξ,τ,α),u¯(ξ,τ,α))} and f¯(ξ,τ,u(ξ,τ),α)=max{f(ξ,τ,v)/v∈(u_(ξ,τ,α),u¯(ξ,τ,α))}.Using Theorems12 and 13 we get the following differential system:(44)∂∂ξLu_ξ,τ,α+asLu_ξ,τ,α=ag_ξ+Lf_ξ,τ,uξ,τ,∂∂ξLu¯ξ,τ,α+asLu¯ξ,τ,α=ag¯ξ+Lf¯ξ,τ,uξ,τsatisfying the following initial conditions:(45)Lu_0,τ,α=Lh_τ,α,Lu¯0,τ,α=Lh¯τ,α.Assume that this leads to(46)Lu_ξ,τ,α=H1s,α,Lu¯ξ,τ,α=K1s,α,where(H1(s,α),K1(s,α)) is solution of system (44) under (45).By the inverse Laplace transform we get(47)u_ξ,τ,α=L-1H1s,α,u¯ξ,τ,α=L-1K1s,α.(b)
Case2: If u is (i)-differentiable with respect to ξ and (ii)-differentiable with respect to τ, then by Theorems 12 and 13 we get the following differential system, satisfying the initial conditions (45):(48)∂∂ξLu_ξ,τ,α+asLu¯ξ,τ,α=ag¯ξ+Lf_ξ,τ,uξ,τ,∂∂ξLu¯ξ,τ,α+asLu_ξ,τ,α=ag_ξ+Lf¯ξ,τ,uξ,τ.Assume that this implies(49)Lu_ξ,τ,α=H2s,α,Lu¯ξ,τ,α=K2s,α,where(H2(s,α),K2(s,α)) is solution of system (48) under (45).Thus(50)u_ξ,τ,α=L-1H2s,α,u¯ξ,τ,α=L-1K2s,α.(c)
Case3: If u is (ii)-differentiable with respect to ξ and (i)-differentiable with respect to τ, then we get the following differential system, satisfying the initial conditions (45):(51)∂∂ξLu¯ξ,τ,α+asLu_ξ,τ,α=ag_ξ+Lf_ξ,τ,uξ,τ,∂∂ξLu_ξ,τ,α+asLu¯ξ,τ,α=ag¯ξ+Lf¯ξ,τ,uξ,τ.Assume that this implies(52)Lu_ξ,τ,α=H3s,α,Lu¯ξ,τ,α=K3s,α,where(H3(s,α),K3(s,α)) is solution of system (51) under (45).Therefore(53)u_ξ,τ,α=L-1H3s,α,u¯ξ,τ,α=L-1K3s,α.(d)
Case4: If u is (ii)-differentiable with respect to ξ and τ, then we get the following differential system, satisfying the initial conditions (45):(54)∂∂ξLu¯ξ,τ,α+asLu¯ξ,τ,α=ag¯ξ+Lf_ξ,τ,uξ,τ,∂∂ξLu_ξ,τ,α+asLu_ξ,τ,α=ag_ξ+Lf¯ξ,τ,uξ,τ.Assume that this leads to(55)Lu_ξ,τ,α=H4s,α,Lu¯ξ,τ,α=K4s,α,where(H4(p,α),K4(p,α)) is solution of system (54) under (45).Hence(56)u_ξ,τ,α=L-1H4s,α,u¯ξ,τ,α=L-1K4s,α.


## 5. Numerical Examples

Example 1.
Consider(57)uξξ,τ=3uτξ,τ+ξ,uξ,0,α=3ξ·α,2-α+ξ22,u0,τ,α=τ·α,2-α,ξ≥0,τ≥0.
Case
1. If u is (i)-differentiable with respect to ξ and τ, then by Laplace transform we get(58)∂∂ξLu_ξ,τ,α=3sLu¯ξ,τ,α-9αξ-3ξ22+ξs,∂∂ξLu¯ξ,τ,α=3sLu_ξ,τ,α+9α-2ξ-3ξ22+ξs.
This differential system satisfies the following initial conditions:(59)Lu_0,τ,α=Lατ=αs2,Lu¯0,τ,α=L2-ατ=2-αs2.
Solving (58) under (59), we get(60)Lu_ξ,τ,α=3αξ+ξ2/2s+αs2,Lu¯ξ,τ,α=6-3αξ+ξ2/2s+2-αs2.
By the inverse Laplace transform we deduce(61)u_ξ,τ,α=α3ξ+τ+ξ22,u¯ξ,τ,α=2-α3ξ+τ+ξ22.
The lengths ofu, uξ, and uτ are, respectively, given by(62)lenuξ,τ,α=u¯ξ,τ,α-u_ξ,τ,α=21-α3ξ+τ≥0,lenuξξ,τ,α=u¯ξξ,τ,α-u_τξ,τ,α=61-α≥0,lenuτξ,τ,α=u¯τξ,τ,α-u_τξ,τ,α=21-α≥0.
So, this solution is valid for allξ≥0 and τ≥0.
Case
2. If u is (i)-differentiable with respect to ξ and (ii)-differentiable with respect to τ, then analogously(63)u_ξ,τ,α=2α-1τ-3ξHτ-3ξ+3αξ+2-ατ+ξ22,u¯ξ,τ,α=21-ατ-3ξHτ-3ξ+32-αξ+ατ+ξ22,where H is the unit step function or the Heaviside function:(64)Hζ=1ζ≥00ζ<0.
Therefore, this solutionu is valid only over Δ={(ξ,τ)∣ξ≥0,τ≥0,τ≤3ξ}.
Case
3. If u is (ii)-differentiable with respect to ξ and (i)-differentiable with respect to τ, then similarly(65)u_ξ,τ,α=2α-1τ-3ξHτ-3ξ+3αξ+2-ατ+ξ22,u¯ξ,τ,α=21-ατ-3ξHτ-3ξ+32-αξ+ατ+ξ22.
As in Case2, one can verify that this solution is valid only over Δ′={(ξ,τ)∣ξ≥0,τ≥0,τ≥3ξ}.
Case
4. If u is (ii)-differentiable with respect to ξ and τ, then(66)u_ξ,τ,α=α3ξ+τ+ξ22,u¯ξ,τ,α=2-α3ξ+τ+ξ22.
One can verify that functionu is not (ii)-differentiable with respect to either ξ or τ.
So, no solution exists in this case.Example 2.
Consider(67)uξξ,τ=uτξ,τ,uξ,0,α=cos⁡ξ·α,2-α,u0,τ,α=cos⁡τ·α,2-α,ξ≥0,τ≥0.
Case
1. If u is (i)-differentiable with respect to ξ and τ, then, by application of the algorithm above, one obtains (68)u_ξ,τ,α=αcos⁡ξ+τ,u¯ξ,τ,α=2-αcos⁡ξ+τ.
The lengths ofu, uξ, and uτ are, respectively, given by(69)lenuξ,τ,α=21-αcos⁡ξ+τ,lenuξξ,τ,α=-21-αsin⁡ξ+τ,lenuξξ,τ,α=-21-αsin⁡ξ+τ.
So, this solution is valid for allξ≥0, τ≥0: ξ+τ∈3π/2+2kπ,2π+2kπ, k∈Z.
Case
2. If u is (i)-differentiable with respect to ξ and (ii)-differentiable with respect to τ, therefore (70)u_ξ,τ,α=α-1cos⁡ξ-τ+cos⁡ξ+τ,u¯ξ,τ,α=1-αcos⁡ξ-τ+cos⁡ξ+τ.
Then, this solution is valid for allξ≥0, τ≥0: ξ-τ∈[3π/2+2kπ,2π+2kπ], k∈Z.
Case
3. If u is (ii)-differentiable with respect to ξ and (i)-differentiable with respect to τ, then analogously(71)u_ξ,τ,α=α-1cos⁡ξ-τ+cos⁡ξ+τ,u¯ξ,τ,α=1-αcos⁡ξ-τ+cos⁡ξ+τ.
Hence, this solution is valid for allξ≥0, τ≥0: ξ-τ∈[2kπ,π/2+2kπ], k∈Z.
Case
4. If u is (ii)-differentiable with respect to ξ and τ, then similarly(72)u_ξ,τ,α=αcos⁡ξ+τ,u¯ξ,τ,α=2-αcos⁡ξ+τ.
So, this solution is valid for allξ≥0, τ≥0: ξ+τ∈[2kπ,π/2+2kπ], k∈Z.


## 6. Conclusion

Theorems of continuity and differentiability for a fuzzy function defined via a fuzzy improper Riemann integral are proved which are used to prove some results concerning fuzzy Laplace transform. Then, using Laplace transform method, the solutions for some linear fuzzy partial differential equations (FPDEs) of first order are given. For future research, one can apply this method to solve FPDEs of high order.



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*Source: 7246027-2016-02-16.xml*