# Hyers-Ulam Stability and Existence Criteria for the Solution of Second-Order Fuzzy Differential Equations

Authors: Noor Jamal; Muhammad Sarwar; M. Motawi Khashan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, existence, uniqueness, and Hyers-Ulam stability for the solution of second-order fuzzy differential equations (FDEs) are studied. To deal a physical model, it is required to insure whether unique solution of the model exists. The natural transform has the speciality to converge to both Laplace and Sumudu transforms only by changing the variables. Therefore, this method plays the rule of checker on the Laplace and Sumudu transforms. We use natural transform to obtain the solution of the proposed FDEs. As applications of the established results, some nontrivial examples are provided to show the authenticity of the presented work.


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## Body

## 1. Introduction

Zadeh [1] introduced the concept of fuzziness in the set theory. The complexity of uncertainty as ambiguity in a real-life scenario is dealt properly with fuzzy set theory. The mathematical tool of fuzzy set theory deals with uncertainty in the real-life problem in a better way. The fuzzy set theory has suggestions for nonclassical and higher order fuzzy sets for different specialized purposes. In this direction, Chang and Zadeh used fuzzy sets and initiate the concept of fuzzy mapping and control [2]. The work of many researchers on fuzzy mappings and control puts the foundation of elementary fuzzy calculus. For detail, see [3–7]. For the last two decades, the valuable interest of fuzzy integral and differential equations has extended classical calculus to modern fuzzy calculus. Fuzzy differential and integral equations are the well-equipped mathematical tools to deal properly with physical models in the fuzzy environments. The solutions of every fuzzy differential and integral equation do not exist. Therefore, some strategy is required to insure whether the solution of FDEs exists or not. The existence theory is one of the best research areas in the field of fuzzy differential equations (FDEs). Before dealing a physical model, it is important to know whether its solution exists. Now, if unique solution of a physical model exists, then a physical model is dealt properly. The existence of a unique solution of fuzzy differential equation properties of differentiable fuzzy mappings was studied by Kaleva [5]. Liu and Liu [8] introduced self-duality credibility measure for the measurement of fuzzy event. Moreover, they studied the existence of unique solutions of FDEs. The existence and uniqueness result for FDEs with linear growth and Lipschitz conditions was discussed by Fei et al. [9]. The uniqueness result for the FDEs with non-Lipschitz coefficients was investigated by Chen and Qin [10] for more detail (see [6, 11, 12]).Stability analysis of differential equations (DEs) is another most important and remarkable area in the qualitative theory. The stability of DEs has been studied by various researchers with different concepts like Lyapunov stability, asymptotic stability, and Ulam stability. The first effort of Ulam stability was initiated by Ulam [13], and just after one year, Hyers [14] studied the stability of the linear functional equation known as Hyers-Ulam stability. Oblaza [15] proved the Hyers-Ulam stability of linear DEs; for further detail of Hyers-Ulam, see [16–19]. Shen [20] investigated the Ulam stability of first-order linear FDEs.The Laplace and Sumudu transforms are commonly used for the solutions of differential equations. The natural transform introduced by Khan and Khan [21] has speciality to converge to both Laplace and Sumudu transforms only by changing the variables. Therefore, the natural transform method plays the rule of checker on the Laplace and Sumudu transforms. The applications of the natural transform method turn out to be well for solutions of differential equations, see [21–23].The aim of this work is to study the existence, uniqueness, and Hyers-Ulam stability of second-order FDEs. For this purpose, the corresponding second-order FDEs are reduced to equivalent systems of fuzzy integral equations. Using the concept of Hukuhara generalized differentiability, existence, uniqueness, and Hyers-Ulam stability of the equivalent system of integral equations are discussed. We use the natural transform method to solve second-order FDEs. The last Hyers-Ulam stability of the numerical problem is discussed. Two nontrivial examples are given to show the authenticity of the presented work.


## 2. Preliminaries

Here, some basic results are provided from the existing literature.Definition 1 (see [24]).
Lets:R→0,1 satisfy the conditions, where y0,y1,y2,z∈R (set of real numbers)
(i)
s is upper semicontinuous(ii)
sσy1+1−σy2≥minsy1,sy2(iii)
∃y0∈R such that σy0=1(iv)
clz∈R,σz>0 is compact
Then,s is a fuzzy number.
Throughout this paper,F^R represent the set that contains all fuzzy numbers.Definition 2 (see [25]).
The fuzzy number can be written ask¯q,k¯q in order pair form, with 0≤q≤1, and holds the following conditions:
(i)
Nondecreasing bounded function,s¯q is left-continuous in 0,1(ii)
Nonincreasing bounded function,s¯q is left-continuous in 0,1(iii)
s¯q≤s¯qq is a crisp nunmber when s¯q=s¯q=q.Theorem 3.
Let the mappingF:Y→Y is contractive with k>1, where Y,d is a generalized complete metric space. If for z∈Y,dFn+1z,Fnz<∞, with n≥0 and Fz0=z0∈Y, then
(i)
Fnz converge to z0(ii)
The pointz0 is in X0=p∈Y∣dFnz,p<∞(iii)
Ifp∈Y, then(1)dp,z0≤11−kdFp,pDefinition 4.
The mappingd:Y1×Y1→R+ is a generalized metric on Y1 if and only if, ∀x,y and z∈Y1.
(F1) dx,y=0 if and only if x=y
(F2) dx,y=dx,y
(F3) dx,z≤dx,y+dy,z. Then, Y1,d is a metric spaceDefinition 5 (see [26]).
The mappingdH:F^R×F^R→R+, defined by
(2)dHx,y=supa1∈0,1dxa1,ya1=supa1∈0,1maxx¯a1−y¯a1,x¯a1−y¯a1.
The pairdH,F^R is a generalized complete metric space. Moreover, ∀x1,x2,x3,x4∈F^R and α∈R+ hold
(D1) dHx1+x3,x2+x3=dHx1,x2
(D2) dHαx1,αx2=αdHx1,x2
(D3) dHx1+x4,x2+x3≤dHx1,x2+dHx4,x3Definition 6 (see [27]).
The fuzzy function,χ:a,b→F^R, at y∈a,b, is generalized Hukuhara differentiable if there exist χ′y∈F^R such that
(i)
Forr>0, sufficiently small, the Hukuhara difference, χy+r⊖χy, χy⊖χy−r, and limits exist in the complete metric space X∈F^R(3)limr→0χy⊖χy+rr=limr→0χy−r⊖χyr=χ′y(ii)
Forr>0, sufficiently small, the Hukuhara difference, χy⊖χy+r, χy−r⊖χy, and limits exist in the complete metric space X∈F^R(4)limr→0χy+r⊖χy−r=limr→0χy⊖χy−r−r=χ′y
The first one is referred to (i)-differentiable and second one to (ii)-differentiable.Definition 7 (see [27]).
The fuzzy function,χ:a,b→F^R, at y∈a,b, is generalized Hukuhara second-order differentiable if there exist, χ″y∈F^R such that
(i)
Forr>0, sufficiently small, the Hukuhara difference, χ′y+r⊖χ′y, χ′y⊖χ′y−r, and limits exist in the complete metric space X∈F^R(5)limr→0χ′y⊖χ′y+rr=limr→0χ′y−r⊖χ′yr=χ″y(ii)
Forr>0, sufficiently small, the Hukuhara difference, χ′y⊖χ′y+r, χ′y−r⊖χ′y, and limits exist in the complete metric space X∈F^R(6)limr→0χ′y+r⊖χ′y−r=limr→0χ′y⊖χ′y−r−r=χ″y
The first one is referred to (i)-differentiable and second one to (ii)-differentiable.Definition 8 (see [28]).
The crisp sete∈Rμe≥γ is the γ-level set of fuzzy number μ∈F^R. Moreover, the γ-level set is bounded and closed with upper and lower bond, μ¯q and μ¯q, respectively, denoted by μ¯q,μ¯q.Theorem 9 (see [29]).
Let a continuous fuzzy functionμ:R→F, such that, μp=μ¯p,μ¯p, and 0≤p≤1:
(i)
μp is (i)-differentiable, then μ′p=μ¯′p,μ¯′p(ii)
μp is (ii)-differentiable, then μ′p=μ¯′p,μ¯′pCorollary 10 (see [26]).
The continuous mappingV:I→F^R is integrable, where I is an interval.Theorem 11 (see [27]).
The generalized differentiable mappingV:I→F^R has integrable derivative V′ on I.
(i)
WhenV is (i)-differentiable, ∀x∈I, we have(7)Vx=Vt0+∫t0x−1V′tdt(ii)
WhenV is (ii)-differentiable, ∀x∈I, we have(8)Vx=Vt0+∫t0x−1V′tdtDefinition 12 (see [21]).
The natural transform ofgt is Rs,p given by the following formula:
(9)Rs,p=Ngt=∫0∞e−stgptdt,
wheres and p are transform parameter which is real and positive.Lemma 13 (see [22]).
The duality relation of natural and Laplace transform is given by(10)Rs,p=Ngt=1p∫0∞e−t/pgtdt=1pLgt,
whereL is the Laplace transform. The natural transform converts to Laplace transform by taking parameter p=1 and Sumudu transform by taking parameter s=1.Definition 14 (see [23]).
The natural transform ofnth order derivative of gt is
(11)Ngnt=snpnRs,p−∑k=0n−1sn−k−1pn−k−11ugk0.
The natural transform ofg′tand g″t first- and second-order derivative of gt is given by
(12)Ng′t=spRs,p−1ug0,Ng″t=s2p2Rs,p−sp2g0−1pg′0.


## 3. Existence, Uniqueness, and Stability Analysis

Before dealing a physical model, it is important to know whether its solution exists. Now, if unique solution of a physical model exists, then the model is dealt properly. But if the solution is not unique then who is to say that the solution we found is what will actually happen in real life. Therefore, we need additional conditions to get unique solution. In this section, we carry out results for the existence of unique solutions of second-order fuzzy differential equations by using the contraction principle.Let us consider FDEs(13)u″t=Ft,ut,u′t,ut0=u0=u¯0,u¯0,u′t0=u0′=u¯0′,u¯0′,where Y=t0,t0+T and F:Y×FR^×FR^⟶FR^, and u0,u0′∈F^R.Definition 15.
The fuzzy-valued function,u∈CY,F^R, is the solution of differential Equation (13) if ut satisfies Equation (13).
Letu∈CY,F^R, which is the solution of differential Equation (13), we reduce Equation (13) to an equivalent system of fuzzy-valued integral equations. Let v:Y→FˆR, which is continuous fuzzy-valued function, such that u’t=vt; therefore, we deduce
(14)u′t=vt,withut0=u0=u¯0,u¯0,v′t=Ft,ut,vt,vt0=u0′=u¯0′,u¯0′.
Using Corollary10, Theorem 11, and initial condition, then the following four systems of equations are obtained:
(1)
Ifu(t) and u′t=vt are (i)-differentiable.(15)ut=u0+∫t0tvsds,vt=u0′+∫t0tFs,us,vsds(2)
Ifu(t) is (i)-differentiable and u′t=vt is (ii)-differentiable.(16)ut=u0+∫t0tvsds,vt=u0′⊖∫t0t−1Fs,us,vsds(3)
Ifu(t) is (ii)-differentiable and u′t=vt is (i)-differentiable.(17)ut=u0⊖∫t0t−1vsds,vt=u0′+∫t0tFs,us,vsds(4)
Ifu(t) and u′t=vt are (ii)-differentiable.(18)ut=u0⊖∫t0t−1vsds,vt=u0′⊖∫t0t−1Fs,us,vsdsTheorem 16.
Let the continuous fuzzy-valued functionu∈CY,F^R and χ=CY,F∧R2 such that H:χ×χ→R+∪+∞, with
(19)Hu1,v1,u2,v2=infN∈R+∪+∞:dHu1,u2+dHv1,v2e−Ct−t0≤Nϕ^t,∀t∈Y,
whereϕ^:Y^→0,∞ be continuous mapping; then, χ,H is a generalized complete metric space.Proof.
The first two conditionsF1 and F2 are easy to show; therefore, we only show F3. Assume that for every u1,v1,u2,v2,u3,v3∈χ, there exist t∈Y, such that
(20)Hu1,v1,u3,v3=dHu1,u3+dHv1,v3e−Ct−t0≤dHu1,u2+dHu2,u3e−Ct−t0⌻+dHv1,v2+dHv2,v3e−Ct−t0=dHu1,u2+dHv1,v2e−Ct−t0⌻+dHu2,u3+dHv2,v3e−Ct−t0,Hu1,v1,u3,v3≤Hu1,v1,u2,v2+Hu2,v2,u3,v3.
Hence,χ,H is a generalized metric space.
Now, we need to show thatχ,H is complete. Let us consider the Cauchy sequence, un,vn on χ,H, then Hun,vn,um,vm≤ε, for all n,m≥Nε, using definition, (19), one can get
(21)dHun,um+dHvn,vme−Ct−t0≤∈ϕ^t,∀n,m≥Nε,t∈Y.
SinceF^R,dH is a complete metric space, then there is ∃ut,vt:Y→F^R, such that Cauchy sequence, unt and vnt converges to ut and vt, respectively.
Whenm→∞, then (21) produced
(22)dHun,ut+dHvn,vte−Ct−t0≤∈ϕ^t,∀n,≥Nε,t∈Y,Hunt,vnt,ut,vt≤εϕ^t,∀n,≥Nε,t∈Y.
Hence,unt,vnt converge to ut,vt, in χ,H. Therefore, χ,H is a complete generalized metric space.Theorem 17.
The fuzzy problem (13) has a unique and Hyers-Ulam stable solution u^,v^:Y→F^R, defined by
(23)u^=u0+∫t0tvsds,v^=u0′+∫t0tFs,us,vsds.
If the following conditions hold for some nonnegativeL,ε and T=t−t0.(i)
dHFt,u1,v1,Ft,u2,v2≤LdHu1,u2+L−1dHv1,v2,
whereF:Y×F^R×F^R→F^R be fuzzy-valued continuous function ∀u1,v1,u2,v2∈F^R and t∈Y(ii)
dHDut,vt+dHDvt,Ft,ut,vt≤ε,
whereu,v:Y→F^R are (i)-differentiable
(iii)
dHu,u^≤1+LεTanddHv,v^≤1+LεTProof.
LetCY,F^R=u:Y→F^R,becontinuousfuzzy−valuedfunction and χ=CY,F∧R2 such that H:χ×χ→R+∪+∞, with
(24)Hu1,v1,u2,v2=infN∈R+∪+∞:dHu1,u2+dHv1,v2e−L+1t−t0≤Nϕ^t∀t∈Y,
whereϕ^:Y^→0,∞ be continuous mapping, then χ,H is a generalized complete metric space, in view of Theorem 16. Let us define a self operator G:χ→χ,∀u,v∈χ and t∈Y, by
(25)Gut,vt=Gu,Gv=u0+∫t0tvsds,u0′+∫t0tFs,us,vsdsds.
First, we showGu,v∈χ. Let p0,q0∈χ and t∈Y,(26)dHGp0,p0+dHGq0,q0e−L+1t−t0<∞,HGp0,q0,p0,q0<∞.
Hence,Gut,vt=Gu,Gv is well defined. Now, we can show for all u,v∈χ, the operator G is strictly contractive on χ,H.(27)dHGu1,Gu2+dHGv1,Gv2=dHu0+∫t0tv1sds,u0+∫t0tv2sds+dHu0′+∫t0tFs,u1s,v1sds,u0′+∫t0tFs,u2s,v2sds=dH∫t0tv1sds,∫t0tv2sds+dH∫t0tFs,u1s,v1sds,∫t0tFs,u2s,v2sds≤L∫t0tdHu1,u2+dHv1,v2e−L+1s−t0eL+1s−t0ds=LL+1Hu1,v1,u2,v2eL+1t−t0.
To multiplye−L+1t−t0, on both sides of the above inequality, we have
(28)dHGu1,Gu2+dHGv1,Gv2e−L+1t−t0≤LL+1Hu1,v1,u2,v2.
Now, from the definition ofH, for all t∈Y, one can get
(29)HGu1,v1,Gu2,v2≤LL+1Hu1,v1,u2,v2.
This indicates thatG is strictly contractive on χ,H. Therefore, there exist unique solution of problem (13).
Theut⊖ut0 and vt⊖u′t0, Hukuhara differences exist for t∈Y, where u and v are i-differentiable, then using condition ii, we have
(30)dHu,Gu+dHv,Gv=dHut,u0+∫t0tvsds+dHvt,u0′+∫t0tFs,us,vsds=dHut−u0,∫t0tvsds+dHvt−u0′,∫t0tFs,us,vsds=dH∫t0tDut,∫t0tvsds+dH∫t0tDvt,∫t0tFs,us,vsds=∫t0tdHDut,vt+dHD2ut,Ft,ut,vtdt≤εt−t0.
To multiplye−L+1t−t0, on both sides of the above inequality, we have
(31)dHu,Gu+dHv,Gve−L+1t−t0≤εt−t0e−L+1t−t0.
Now, from the definition ofH and substitute T=t−t0, for all t∈Y, we can obtain
(32)Hu,v,Gu,v≤εTe−L+1t−t0.
From, Theorem3 condition iii, there exist unique solution u^,v^, of problem (13), such that
(33)Hu,v,u^,v^=11−Hu,v,Gu,v≤1+LεTe−L+1t−t0.
Using definition ofH, one can get
(34)dHu,u^+dHv,v^≤1+LεT,∀t∈Y.
Hence, for allt∈Y, we can get
(35)dHu,u^≤1+LεT,dHv,v^≤1+LεT.
Hence, the fuzzy solutions of the problem (13) are Hyers-Ulam stable.Theorem 18.
LetF:Y×F^R×F^R→F^R be the fuzzy-valued continuous function such that
(36)dHFt,u1,v1,Ft,u2,v2≤dHu1,u2,∀u1,v1,u2,v2∈F^R,t∈Y.
If the functions,u:Y→F^R, are (i)-differentiable and v:Y→F^R is (ii)-differentiable, the following condition holds:
(37)dHDut,vt+dHDvt,Ft,ut,vt≤ε,somenonnegative,ε.
Then, there exist unique solution of the problem (13), u^,v^:Y→F^R, defined by
(38)u^=u0+∫t0tvsds,v^=u0′−1⊝∫t0tFs,us,vsds.
The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied:(39)dHu,u^≤L+1εT,dHv,v^≤L+1εT,
whereL is nonnegative constant and T=t−t0.Proof.
The proof can be easily obtained on the similar procedure of Theorem17.Theorem 19.
LetF:Y×F^R×F^R→F^R, be the fuzzy-valued continuous function such that
(40)dHFt,u1,v1,Ft,u2,v2≤dHu1,u2,∀u1,v1,u2,v2∈F^R,t∈Y.
If the functionu:Y→F^R is (ii)-differentiable and v:Y→F^R is (i)-differentiable, the following condition holds:
(41)dHDut,vt+dHDvt,Ft,ut,vt≤ε,somenonnegative,ε.
Then, there exist unique solutions of the problem (13), u^,v^:Y→F^R, defined by
(42)u^=u0−1⊝∫t0tvsds,v^=u0′+∫t0tFs,us,vsds.
The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied:(43)dHu,u^≤L+1εT,dHv,v^≤L+1εT,
whereL is nonnegative constant and T=t−t0.Proof.
The proof can be easily obtained on the similar procedure of Theorem17.Theorem 20.
LetF:Y×F^R×F^R→F^R be the fuzzy-valued continuous function such that
(44)dHFt,u1,v1,Ft,u2,v2≤dHu1,u2,∀,u1,v1,u2,v2∈F^R,t∈Y.
If the functionsu,v:Y→F^R are (ii)-differentiable, the following conditions hold:
(45)dHDut,vt+dHDvt,Ft,ut,vt≤ε,somenonnegative,ε.
Then, there exist unique solution of the problem (13), u^,v^:Y→F^R, defined by
(46)u^=u0+∫t0tvsds,v^=u0′+∫t0tFs,us,vsds.
The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied:(47)dHu,u^≤L+1εT,dHv,v^≤L+1εT,
whereL is nonnegative constant and T=t−t0.Proof.
Using a procedure similar to Theorem17, one can easily prove.


## 4. Natural Transforms and Solutions of the Proposed Fuzzy Differential Equations

Theorem 21.
Let fuzzy-valued functionχw be continuous such that e−swχuw, e−swχ′uw, and e−swχ′′uw exist and are continuous. Moreover, they are Riemann integrable on 0,∞; then,
(A)
Ifχw and χ′w are (i)-differentiable, then(48)Nχ′′w=s2u2R^s,u⊖su2χ0⊖1uχ′0(B)
Ifχw is (i)-differentiable and χ′w is (ii)-differentiable, then(49)Nχ′′w=−1uχ′0⊖−s2u2R^s,u⊖−su2χ0(C)
Ifχw is (ii)-differentiable and χ′w is (i)-differentiable, then(50)Nχ′′w=−su2χ0⊖−s2u2R^s,u⊖1uχ′0.(D)
Ifχw and χ′w are (ii)-differentiable, then(51)Nχ′′w=−1uχ′0⊖su2χ0⊖s2u2R^s,uProof.
(A)
Letχw and χ′w are (i)-differentiable, then(52)Nχ′w=suR^s,u⊖1uχ0,andNχ′′w=suNχ′w⊖1uχ′0.
Using (52), this identity yields
(53)Nχ′′w=susuR^s,u⊖1uχ0⊖1uχ′0,Nχ′′w=s2u2R^s,u⊖su2χ0⊖1uχ′0(B)
If,χw and χ′w are (i)-differentiable and (ii)-differentiable, respectively, then(54)Nχ′w=suR^s,u⊖1uχ0,Nχ′′w=−1uχ′0⊖−suNχ′w.
Using (54), this identity yields
(55)Nχ′′w=−1uχ′0⊖−susuR^s,u⊖1uχ0,Nχ′′w=−1uχ′0⊖−s2u2R^s,u⊖−su2χ0(C)
Ifχ′w and χw are (i)-differentiable and (ii)-differentiable, respectively, then(56)Nχ′w=−1uχ0⊖−suR^s,u,Nχ′′w=suNχ′w⊖1uχ0.
Using (56), this identity yields
(57)Nχ′′w=−su2χ′0⊖−s2u2R^s,u⊖1uχ0,Nχ′′w=−1uχ0⊖−s2u2R^s,u⊖1uχ′0(D)
Ifχw and χ′w are (ii)-differentiable, then(58)Nχ′w=−1uχ0!−suR^s,u,Nχ′′w=−1uχ′0⊖−suNχ′w.
Using (58), this identity yields
(59)Nχ′′w=−1uχ′0⊖−su−1uχ′0⊖−suR^s,u,Nχ′′w=−1uχ′0⊖su2χ0⊖s2u2R^s,uNext, we have to solve problem (13), with natural transform. The fuzzy solution of the concerned problem can be discussed in four cases. Taking natural transform of (13), we can get
(60)NFw,χw,χ′w=Nχ′′w.Case I.
Letχw and χ′w are (i)-differentiable; then, using Theorem 21, in (60), one can get
(61)NF¯w,χw,χ′w,γ=s2u2R^¯s,u⊖su2χ¯0,γ⊖1uχ¯′0,γ,NF¯w,χw,χ′w,γ=s2u2R^¯s,u⊖su2χ¯0,γ⊖1uχ¯′0,γ,
where(62)F¯w,χw,χ′w,γ=minFw,χw,χ′w∣χw∈χ¯w,γ,χ¯w,γ;χ′w∈χ¯′w,γ,χ¯′w,γ,F¯w,χw,χ′w,γ=maxFw,χw,χ′w∣χw∈χ¯w,γ,χ¯w,γ;χ′w∈χ¯′w,γ,χ¯′w,γ.
Assume that solving Equation (61), satisfying the conditions χw0=χ0=χ¯0,χ¯0 and χ′w0=χ0′=χ¯0′,χ¯0′, one can obtain the solutions as follows:
(63)R^¯s,u,γ=R1s,u,γ,R¯^s,u,γ=G1s,u,γ.
Taking inverse fuzzy natural transform of Equation (63), one can find χ¯w,γ and χ¯w,γ, as follows:
(64)χ¯w,γ=N−1R1s,u,γ,χ¯w,γ=N−1G1s,u,γ.Case II.
Letχw is (i)-differentiable and χ′w is (ii)-differentiable, then using Theorem 21, in (60), one can get
(65)NF¯w,χw,χ′w,γ=−1uχ¯′0,γ⊖−s2u2R^¯s,u,γ⊖−su2χ¯0,γ,NF¯w,χw,χ′w,γ=−1uχ¯′0,γ⊖−s2u2R¯^s,u,γ⊖−su2χ¯0,γ.
Using initial condition in (65), χw0=χ0=χ¯0,χ¯0 and χ′w0=χ0′=χ¯0′,χ¯0′, one can get
(66)NF¯w,χw,χ′w,γ=s2u2R^¯s,u,γ⊖su2χ¯0γ⊖1uχ¯0′γ,NF¯w,χw,χ′w,γ=s2u2R¯^s,u,γ⊖su2χ¯0γ⊖1uχ¯0′γ.
Assume that by solving Equation (66), we can get the solutions as follows:
(67)R^¯s,u,γ=R2s,u,γ,R¯^s,u,γ=G2s,u,γ.
Taking inverse fuzzy natural transform of Equation (67), one can find χ¯w,γ and χ¯w,γ,(68)χ¯w,γ=N−1R2s,u,γ,χ¯w,γ=N−1G2s,u,γ.Case III.
Letχw is (ii)-differentiable and χ′w is (i)-differentiable, then using Theorem 21, in (60), one can get
(69)NF¯w,χw,χ′w,γ=−su2χ¯0,γ⊖−s2u2R^¯s,u,γ⊖−1uχ¯′0,γ,NF¯w,χw,χ′w,γ=−su2χ¯0,γ⊖−s2u2R^¯s,u,γ⊖−1uχ¯′0,γ.
Using initial condition in (69), χw0=χ0=χ¯0,χ¯0 and χ′w0=χ0′=χ¯0′,χ¯0′, one can get
(70)NF¯w,χw,χ′w,γ=s2u2R^¯s,u,γ⊖su2χ¯0γ⊖1uχ¯0′γ,NF¯w,χw,χ′w,γ=s2u2R¯^s,u,γ⊖su2χ¯0γ⊖1uχ¯0′γ.
Assume that by solving Equation (70), one can obtain the solution as follows:
(71)R^¯s,u,γ=R3s,u,γ,R¯^s,u,γ=G3s,u,γ.
Taking inverse fuzzy natural transform of Equation (71), one can find
(72)χ¯w,γ=N−1R3s,u,γ,χ¯w,γ=N−1G3s,u,γ.Case IV.
Letχw and χ′w are (ii)-differentiable, then using Theorem 21, in (60), one can get
(73)NF¯w,χw,χ′w,γ=−su2χ¯0,γ⊖−s2u2R^¯s,u,γ⊖−1uχ¯′0,γ,NF¯w,χw,χ′w,γ=−su2χ¯0,γ⊖−s2u2R^¯s,u,γ⊖−1uχ¯′0,γ.
Using initial condition in (73), χw0=χ0=χ¯0,χ¯0 and χ′w0=χ0′=χ¯0′,χ¯0′, one can get
(74)NF¯w,χw,χ′w,γ=s2u2R^¯s,u,γ⊖su2χ¯0γ⊖1uχ¯0′γ,NF¯w,χw,χ′w,γ=s2u2R¯^s,u,γ⊖su2χ¯0γ⊖1uχ¯0′γ.
Assume that by solving Equation (74), one can obtain the solution as follows:
(75)R^¯s,u,γ=R4s,u,γ,R¯^s,u,γ=G4s,u,γ.
Taking inverse fuzzy natural transform of Equation (75), one can find
(76)χ¯w,γ=N−1R4s,u,γ,χ¯w,γ=N−1G4s,u,γ.


## 5. Examples

Here, we discuss the numerical problem.(77)χ′′w=pwχw+qwχ′w+cw,χw0=χ0=χ¯0,χ¯0,χ′w0χ0′=χ¯0′,χ¯0′,wherep,q:Y→R+ be continuous functions such that for nonnegative L,pw≤L,qw≤L−1, and c:Y→F^R be continuous function for all w∈Y, while Y=0,3/L. If the functions χ:Y→F^R, the following conditions for some nonnegative ε hold:
(78)dHDχw,χ′w+dHDχ′w,pwχw+qwχ′w+cw≤ε.Assume that, on settingFw,χw,χ′w=pwχw+qwχ′w+ww, it is easy to obtain condition (i), as follows
(79)dHFw,χ1,χ1′,Fw,χ2,χ2′=dHpwχ1w+qwχ1′w+cw,pwχ2w+qwχ2′w+cw≤dHpwχ1w,pwχ2w+dHqwχ1′w,qwχ2′w=∣pw∣dHχ1w,χ2w+∣qw∣dHχ1′w,χ2′w≤LdHχ1w,χ2w+L−1dHχ1′w,χ2′w,forallw∈YThen, there exist unique solution of the problem (77), u^,v^:Y→F^R, defined by
(80)χ^=χ0+∫3/2Lwχ′sds,χ^′=χ0′+∫3/2LwFs,χs,χ′sds,(81)dHχ,χ^≤1+L32Lε,dHχ′,χ^′≤1+L32Lε,forall,w∈Y.This shows that the fuzzy solution of differential equations, (77), is Hyers-Ulam stable on Y.Whenpw=p and qw=q, then R1s,u,γ and G1s,u,γ of system (60) can be calculated asCase 1.
Whenp,q≥0, then system (61) is equivalent to
(82)s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=p+qsuR¯^s,u,γ⊖q1uχ¯′0,γ+Nc¯w,s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=p+qsuR¯^s,u,γ⊖q1uχ¯′0,γ+Nc¯w.
After some process, one can get(83)R1s,u,γ=R¯^s,u,γ=s/u2−q/uχ¯0,γ+1/uχ¯′0,γ+Nc¯ws2/u2−qs/u−p,G1s,u,γ=R¯^s,u,γ=s/u2−q/uχ¯0,γ+1/uχ¯′0,γ+Nc¯ws2/u2−qs/u−p.Case 2.
Whenp≥0 and q<0, then system (65) is equivalent to the following:
(84)s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=pR¯^s,u,γ+qsuR¯^s,u,γ⊖quχ¯′0,γ+Nc¯w,s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=pR¯^s,u,γ+qsuR¯^s,u,γ⊖quχ¯′0,γ+Nc¯w.
Let us denote(85)fs,u,γ=su2χ¯0,γ+1uχ¯′0,γ−quχ¯′0,γ+Nc¯w,gs,u,γ=su2χ¯0,γ+1uχ¯′0,γ−quχ¯′0,γ+Nc¯w.
Solving the above system, one can get(86)R1s,u,γ=R¯^s,u,γ=s2/u2−pfs,u,γ+qs/ugs,u,γs2/u2−p2+qs/u2,G1s,u,γ=R¯^s,u,γ=s2/u2−pgs,u,γ−qs/ufs,u,γs2/u2−p2+qs/u2.Case 3.
Whenp<0 and q≥0, then system (69) is equivalent to the following system:
(87)s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=pR¯^s,u,γ+qsuR¯^s,u,γ⊖quχ¯′0,γ+Nc¯w,s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=pR¯^s,u,γ+qsuR¯^s,u,γ⊖quχ¯′0,γ+Nc¯w.
Let us denote(88)Bs,u,γ=su2χ¯0,γ+1uχ¯′0,γ−quχ¯′0,γ+Nc¯w,Cs,u,γ=su2χ¯0,γ+1uχ¯′0,γ−quχ¯′0,γ+Nc¯w.
Solving the above system, one can get(89)R1s,u,γ=R¯^s,u,γ=s2/u2−qs/uBs,u,γ+pCs,u,γs2/u2−qs/u2+p2,G1s,u,γ=R¯^s,u,γ=s2/u2−qs/uCs,u,γ+pBs,u,γs2/u2−qs/u2+p2.Case 4.
Whenp<0 and q<0, then system (73) is equivalent to the following:
(90)s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=pR¯^s,u,γ+qsuR¯^s,u,γ⊖quχ¯′0,γ+Nc¯w,s2u2R¯^s,u,γ⊖su2χ¯0,γ⊖1uχ¯′0,γ=pR¯^s,u,γ+qsuR¯^s,u,γ⊖quχ¯′0,γ+Nc¯w.
For the remaining system, we can find the same processes. Solving the above system, one can get(91)R1s,u,γ=R¯^s,u,γ=s2/u2fs,u,γ+p+qs/ugs,u,γs2/u22+p+qs/u2,G1s,u,γ=R¯^s,u,γ=s2/u2gs,u,γ+p+qs/ufs,u,γs2/u22+p+qs/u2.Example 1.
Now, we discuss the following system:(92)χ′′t=−χt+w0,w0=γ,2−γ,χ0,γ=γ−1,1−γ,χ′0,γ=γ−1,1−γ.
This differential equation is equivalent to Case3 of problem (77); therefore, the solution exists.
(93)s2u2R¯^s,u,γ⊖su2γ−1⊖1uγ−1=−R¯^s,u,γ+Nw¯t,s2u2R¯^s,u,γ⊖su21−γ⊖1u1−γ=−R¯^s,u,γ+Nw¯t.
Solving the above system, one can get(94)R1s,u,γ=R¯^s,u,γ=γ12s+u−12s−u+1s+12s−u−12s+u−ss2+u2,G1s,u,γ=R¯^s,u,γ=γ12s−u−12s+u−1s+2s+12s+u−12s−u−ss2+u2.
Now, applying inverse natural transform, we deduce(95)χ¯t,γ=γ1−sinht+sinht−cost,χ¯t,γ=2−γ1−sinht+sinht−cost.
This solution is acceptable for allt∈0,ln1+√2, see [30] (Figure 1).Figure 1
Present fuzzy solutions of Example1 for different values of γ. The subplot (i) is 2D plot of lower and upper solutions with t=0.1, and subplot (ii) is 2D plot of lower and upper solutions with t=0.3.Example 2.
Here, we consider the following system:(96)χ′′t=χt+t+1,χ0,γ=γ−2,1−2γ,χ′0,γ=γ−2,1−2γ.
This differential equation is equivalent to Case1 of problem (77); therefore, the solution exists.
(97)s2u2R¯^s,u,γ⊖su2γ−2⊖1uγ−2=−R¯^s,u,γ+us2+1s,s2u2R¯^s,u,γ⊖su21−γ⊖1u1−γ=−R¯^s,u,γ+us2+1s.
Solving Equation (97), one can write
(98)R1s,u,γ=R¯^s,u,γ=γs−u−2s−u+1s2s−u+1s3s−u,G1s,u,γ=R¯^s,u,γ=1s−u−2γs−u+1s2s−u+1s3s−u.
Now, applying inverse natural transform, we deduce(99)χ¯t,γ=γexpt−t22−2t−2,χ¯t,γ=3−2γexpt−t22−2t−2.
This solution is acceptable for allt∈R (real numbers) (Figure 2).Figure 2
Present fuzzy solutions of Example2 for different values of γ. The subplot (i) is 2D plot of lower and upper solutions with t=5, and subplot (ii) is 2D plot of lower and upper solutions with t=0.5.


## 6. Conclusion and Future Direction

In this work, we study the existence, uniqueness, and Hyers-Ulam stability of second-order FDEs. We discuss the natural transform of second-order fuzzy differential equations for different cases of fuzzy differentiability. The natural transform converts to Laplace transform by taking parameterp=1 and Sumudu transform by taking parameter s=1. Therefore, we use natural transform to solve the proposed FDEs. In the future, the natural transform method will be used to solve fuzzy problems due to significant duality with Laplace and Sumudu transforms.



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