Int. J. Anal. Appl. (2023), 21:76 An Algorithm for the Solution of Second Order Linear Fuzzy System With Mechanical Applications S. Nagalakshmi1,∗, G. Suresh Kumar1, Ravi P. Agarwal2, Chao Wang3 1Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, 522302, Andhra Pradesh, India 2Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USA 3Department of Mathematics, Yunnan University, Kunming 650091, China ∗Corresponding author: nagalakshmi.soma@gmail.com Abstract. In this paper, we consider homogeneous and non-homogeneous second order linear fuzzy systems under granular differentiability. The concept of continuous n-dimensional fuzzy functions on the space of n-dimensional fuzzy numbers are introduced. Developed an algorithm for the solution of a non-homogeneous second order linear fuzzy system under granular differentiability. The proposed algorithm is applied to solve some well-known mechanical problems with fuzzy uncertainty. 1. Introduction Mathematical models can be explained through fuzzy differential equations (FDE). The innovative work on system of fuzzy differential equations (SFDEs) extended from population models, bio infor- matics, quantum optics, and soft computing models. Second-order linear fuzzy systems (SLFS) are modeled by behaviors of many dynamical systems with uncertainty. SLFSs specifically appear in many spring-mass mechanical systems with uncertainty. Fard and Ghal-EH [3] proposed a numerical method to solve SFDEs under H-differentiability. Gasilov et al. [4] presented a solution method for SFDEs with fuzzy initial conditions. Mondal et al. [7] analyzed adaptive schemes to study the SFDEs. Barazandeh and Ghazanfari [1] obtained the solutions for SFDEs applying variation iteration technique. Keshavarz et al. [5] enhanced to obtain an analytical solution for SFDEs using gH-differentiability. Boukezzoula Received: Apr. 2, 2023. 2020 Mathematics Subject Classification. 34A07, 34B05. Key words and phrases. n-dimensional fuzzy number; second-order granular derivative; system of fuzzy initial value problems. https://doi.org/10.28924/2291-8639-21-2023-76 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-76 2 Int. J. Anal. Appl. (2023), 21:76 et al. [2] enhanced a method to solve the SFDEs with variables as fuzzy intervals. The limitations of previous methods for dealing with SFDEs are derivatives do not always exist, monotonicity of the uncertainty, doubling properties, unnatural behavior in modeling phenomenon, and multiplicity of solutions. Piegat and Landowski [11] introduced horizontal membership function (HMF), and their applica- tions. Piegat and Pluciński [12] was stated the difference between relative distance measure interval arithmetic (RDM-IA) yields a multidimensional answer while the results produced with SIA. Mazan- darani et al. [6] elaborated the concept of HMF, granular differentiability (gr-differentiability) and granular integrability (gr-integrability). Najariyan and Zhao [9] offered a solution to the fuzzy dynam- ical system under gr-differentiability. Nagalakshmi et al. [8] generalized the concept of fuzzy numbers to n-dimensional fuzzy numbers and developed an algorithm to solve system of first-order FBVPs under the concept of gr-differentability. In this manuscript, consider two types of SLFSs under gr-differentiability. The upcoming sections of this manuscript are along these lines. Section 2, presents basic definitions and propositions related to gr-differentiability of n-dimensional fuzzy valued function. Section 3, an algorithm is presented as a working method to solve SLFSs under gr-differentiability. In Section 4, we describe mechanical applications such as automobile two-axles, railway cars system, and spring-mass systems to highlight the proposed algorithm. Section 5, Conclusions and future works are analyzed. 2. Preliminaries For a later discussion, this section provides some essential notations, definitions, and findings. Suppose that the membership function, q : R → [0,1] of a fuzzy subset of the real number set R, satisfies the following conditions: (i) q(t0)=1 for at least one t0 ∈R. (ii) q(λy +(1−λ)z)≥ min{q(y),q(z)},∀λ ∈ [0,1], y,z ∈R. (iii) q is upper semi continuous on R. (iv) cl{t ∈R;q(t) > 0} is compact. Then it is called a fuzzy number (FN). Here q(t) is the membership degree of t, ∀t ∈ R. The λ-level sets of q are defined by [q]λ = {t ∈ R : q(t) ≥ λ} = [qλl ,q λ r ], for 0 < λ ≤ 1 and [q]0 = cl{t ∈R : q(t) > 0}. Let RF denotes the space of FNs in R. Refer to [6] for definitions, notations, and essential findings regarding HMFs, first-order granular derivative (gr-derivative), and granular integration (gr-integrations) of FNs in R. Definition 2.1. Suppose that p,q ∈ RF, whose HMFs are pgr(λ,αp) and qgr(λ,αq) respectively. Then r = p ∗ q ∈ RF, such that H(r) , pgr(λ,αp)o qgr(λ,αq), where “o” and “∗” denotes any one of the operations addition, multiplication, subtraction and division in R and RF, respectively and 0 /∈ qgr(λ,αq) if “∗” denotes the division. That is Int. J. Anal. Appl. (2023), 21:76 3 (1) H(p⊕q), pgr(λ,αp)+qgr(λ,αq), (2) H(p⊗q), pgr(λ,αp)qgr(λ,αq), (3) H(p q), pgr(λ,αp)−qgr(λ,αq), (4) H(p�q), pgr(λ,αp)÷qgr(λ,αq), (5) H(k �q), k qgr(λ,αq), where k ∈R and p,q,r ∈RF. Definition 2.2. [9] Let f : [a,b]→RF, be the FF. If there exists d2grf (t0) dt2 ∈RF, such that lim h→0 f ′(t0 +h) f ′(t0) h = d2grf (t0) dt2 = f ′′gr(t0), then f is said to be second order gr-differentiable at a point t0 ∈ [a,b]. Theorem 2.1. [9] Let f : [a,b]→RF. Then f is twice gr-differentiable if and only if its HMF is twice differentiable with respect to t ∈ [a,b]. Moreover, H ( d2grf (t) dt2 ) = ∂2fgr(t,λ,αf ) ∂t2 . Proposition 2.1. Let f : [a,b] → RF be a FF, with [f (t)]λ = [ fλl (t), f λ r (t) ] . The FF f is gr- differentiable twice on [a,b] if and only if (fλl ) ′(t) and (fλr ) ′(t) are differentiable on [a,b]. Proof. Since [f (t)]λ = [ fλl (t), f λ r (t) ] , then fgr(t,λ,αg)= fλl (t)+(f λ r (t)−fλl (t))αf , where λ, αf ∈ [0,1]. From Definition 2.2 and Theorem 2.1, we have Suppose that f (t) is a gr-differentiable twice on [a,b] ⇐⇒ ∂2fgr(t,λ,αf ) ∂t2 =(fλl ) ′′(t)+((fλr ) ′′(t)− (fλl ) ′′(t))αf ⇐⇒ (fλl ) ′(t) and (fλr ) ′(t) are differentiable on [a,b]. . � Definition 2.3. [8] Let RnF =RF ×RF ×RF ×···×RF︸ ︷︷ ︸ ntimes , be the space of n-dimensional fuzzy vectors whose components are fuzzy numbers. Then the addition and scalar multiplication defined component wise as follows: If u =(u1,u2, · · · ,un),v =(v1,v2, · · · ,vn)∈RnF, then (i) u ⊕v =(u1 ⊕v1,u2 ⊕v2, · · · ,un ⊕vn), (ii) k �u =(k �u1,k �u2, · · · ,k �un), where ui,vi ∈RnF, i =1,2, · · · ,n and k ∈R. Definition 2.4. If u =(u1,u2, · · · ,un)∈RnF, as ui ∈RF, i =1,2, · · · ,n. Then the HMF for u ∈R n F is defined by ugr(λ,αu)= (u1gr(λ,α1),u2gr(λ,α2), · · · , ungr(λ,αn)), where λ, α1, · · · ,αn ∈ [0,1]. 4 Int. J. Anal. Appl. (2023), 21:76 Proposition 2.2. Let u and v be two n-dimensional fuzzy vectors. Then u and v are said to be equal if and only if H(u)= H(v), for all αu = αv ∈ [0,1]. Proof. Since u,v ∈ RnF, then u = (u1,u2, · · · ,un),v = (v1,v2, · · · ,vn), for ui, vi ∈ RF, i = 1,2, · · · ,n. Consider, u = v ⇐⇒ (u1,u2, · · · ,un)= (v1,v2, · · · ,vn) ⇐⇒ ui = vi, i =1,2, · · · ,n. ⇐⇒ H(ui)= H(vi), for all αui = αvi ∈ [0,1], i =1,2, · · · ,n. ⇐⇒ (H(u1),H(u2), · · · ,H(un))= (H(v1),H(v2), · · · ,H(vn)) ⇐⇒ H(u)= H(v), for all αu = αv ∈ [0,1], where αu , (αu1,αu2, · · · ,αun) and αv , (αv1,αv2, · · · ,αvn). � Definition 2.5. [8] Let u,v ∈RnF. The function D n gr :R n F ×R n F → R + ∪{0}, defined by Dngr(u,v)= sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖, which is called a n-dimensional granular distance between two n-dimensional fuzzy vectors u and v, where ‖.‖ represents Euclidean norm in Rn. Proposition 2.3. The function Dngr is a metric on the space of RnF. Proof. Suppose that RnF is a non-empty set and D n gr :R n F ×R n F → R + ∪{0} is real-valued function. (i) Consider, Dngr(u,v)= sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖ > 0. (ii) Consider, Dngr(u,v)=0 ⇐⇒ sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖=0 ⇐⇒ ‖ugr −vgr‖=0 ⇐⇒ ugr −vgr =0 ⇐⇒ ugr = vgr ⇐⇒ u = v. (iii) Consider, Dngr(u,v)= sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖ =sup λ max αu,αv ‖vgr(λ,αv)−ugr(λ,αu)‖ =Dngr(v,u). Int. J. Anal. Appl. (2023), 21:76 5 (iv) Consider, Dngr(u,w)= sup λ max αu,αw ‖ugr(λ,αu)−wgr(λ,αw)‖ =sup λ max αu,αv,αw ‖ugr(λ,αu)−vgr(λ,αv)+vgr(λ,αv)−wgr(λ,αw)‖ ≤ sup λ max αu,αv ‖ugr(λ,αu)gr −vgr(λ,αv)‖+sup λ max αv,αw ‖vgr(λ,αv)−wgr(λ,αw)‖ =Dngr(u,v)+D n gr(v,w). From (i)-(iv), ( RnF,D n gr ) is a metric space. � Theorem 2.2. ( RnF,D n gr ) is a complete metric space (CMS). Proof. If any Cauchy sequence of n-dimensional fuzzy vectors in ( RnF,D n gr ) is convergent then the proof concluded. Suppose that um ∈ RnF, m ≥ 1 is a Cauchy sequence. Then for all �1 > 0, there exists N ≥ 1 such that Dngr(um,um+p) < �1, for all m ≥ 1, q ≥ 1. Dngr(um,um+p) < �1 =⇒ sup λ max αum,αum+p ‖umgr(λ,αum)−um+pgr(λ,αum+p)‖ < �1 =⇒ ‖umgr −um+pgr‖ < �1. Now {umgr} is a Cauchy sequence in the space of Rn. Clearly { umgr } is convergent in Rn and umigr(λ,αumi)= umi λ l +(u λ mir −uλmil)αumi , where λ,αumi ∈ [0,1]. Since umigr(λ,αumi) is convergent, so that u λ mil and uλmir are convergent. Suppose that lim n→∞ uλmil = ui λ l and limn→∞ uλmir = ui λ r . Since u λ mil ≤ uλmir, so that u λ i l ≤ uλi r for all i = 1,2, · · · ,n. If [uλi l,u λ i r ], i = 1,2, · · · ,n are λ-level sets of ui, then proof will be complete. It is shown in the same manner in the proof of Theorem 4 [6], and therefore is left off. � Lemma 2.1. Suppose that u,v,w,s ∈RnF and µ ∈R, then the below results hold: (i) Dngr(u ⊕v,w ⊕ s)≤Dngr(u,w)+Dngr(v,s). (ii) Dngr(µ�u,µ�v)= |µ|Dngr(u,v). (iii) Dngr(u ⊕v,w ⊕v)≤Dngr(u,w). Proof. (i) From Definition 2.5, we have Dngr(u ⊕v,w ⊕ s) = sup λ max αu,αv,αw,αs ‖(ugr(λ,αu)+vgr(λ,αv))− (wgr(λ,αw)+ sgr(λ,αs))‖ =sup λ max αu,αv,αw,αs ‖(ugr(λ,αu)−wgr(λ,αw))+(vgr(λ,αv)− sgr(λ,αs))‖ 6 Int. J. Anal. Appl. (2023), 21:76 ≤ sup λ max αu,αw ‖ugr(λ,αu)−wgr(λ,αw)|+sup λ max αv,αs |(vgr(λ,αv)− sgr(λ,αs))‖ =Dngr(u,w)+D n gr(v,s). (ii) From Definition 2.5, we have Dngr(µ�u,µ�v)= sup λ max αu,αv,αµ ‖µugr(λ,αu)−µvgr(λ,αv)‖ = |µ| sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖ = |µ|Dngr(u,v). (iii) From (i), we have Dngr(u ⊕v,w ⊕v) =Dngr(u,w)+D n gr(v,v) =Dngr(u,w). � Proposition 2.4. If f : [a,b]→RnF is a fuzzy function, then it is called an n-dimensional fuzzy valued function on [a,b]. Proof. Since f : [a,b]→RnF is a fuzzy function, then f (t)∈R n F, for all t ∈ [a,b]. Therefore f (t)= (f1(t), f2(t), . . . , fn(t)), for all t ∈ [a,b] and fi(t)∈RF, i =1,2, . . . ,n. Thus f (t) is a n-dimensional fuzzy vector for each t ∈ [a,b] and hence f : [a,b] → RnF, is a n- dimensional fuzzy valued function on [a,b]. � Proposition 2.5. If f : [a,b]→RnF is a n-dimensional fuzzy valued function, include mn ∈ N distinct FNs, then the HMF of f is denoted by H(f (t)), fgr(t,λ,αf ), and interpreted as fgr : [a,b]×[0,1]× [0,1]× [0,1]×···× [0,1]︸ ︷︷ ︸ mntimes → Rn, in which αf , (αi1,αi2, . . . ,αim), where αi1,αi2, . . . ,αim are the mn RDM variables for ui1, ui2, . . ., uim for i =1,2, · · · ,n. Proof. Since f : [a,b] → RnF is a n-dimensional fuzzy valued function, so that f (t) = (f1(t), f2(t), . . . , fn(t)), for all t ∈ [a,b] and fi(t)∈RnF, i =1,2, . . . ,n. Therefore H(f (t))= (H(f1(t)),H(f2(t)), . . . ,(fn(t))) fgr(t,λ,αf )= (f1gr(t,λ,α11,α12, . . . ,α1m), f2gr(t,λ,α21,α22, . . . ,α2m), . . . , fngr(t,λ,αn1,αn2, . . . ,αnm)) where αf ≡ (αi1,αi2, . . . ,αim)∈ [0,1], i =1,2, · · · ,n. � Int. J. Anal. Appl. (2023), 21:76 7 Definition 2.6. Let f : [a,b] → RnF be a n-dimensional fuzzy valued function. The limit of f (t) as t → p is q ∈RnF, which is subject to following conditions: (i) If p ∈ (a,b), for all �1 > 0, there exits δ1 > 0 such that |t−p| < δ1 =⇒ Dngr(f (t),q) < �1, and write it as lim t→p f (t)= q. (ii) If p = b, for all �1 > 0, there exits δ1 > 0 such that 0 < t −b < δ1 =⇒ Dngr(f (t),q) < �1, and write it as lim t→b+ f (t)= q. (iii) If p = c, for all �1 > 0, there exits δ1 > 0 such that 0 < c − t < δ1 =⇒ Dngr(f (t),q) < �1, and write it as lim t→c− f (t)= q. Definition 2.7. Let f : [a,b] → RnF be a n-dimensional fuzzy valued function. The function f (t) is said to be continuous at t = p if f (p)∈RnF, which is subject to following conditions: (i) If p ∈ (a,b), for all �1 > 0, there exits δ1 > 0 such that |t−p| < δ1 =⇒ Dngr(f (t), f (p)) < �1, and write it as lim t→p f (t)= f (p). (ii) If p = b, for all �1 > 0, there exits δ1 > 0 such that0 < t−b < δ1 =⇒ Dngr(f (t), f (b)) < �1, and write it as lim t→b+ f (t)= f (b). (iii) If p = c, for all �1 > 0, there exits δ1 > 0 such that0 < c−t < δ1 =⇒ Dngr(f (t), f (c)) < �1, and write it as lim t→c− f (t)= f (c). Note 2.1. [8] If f ,h : [a,b]→RnF are n-dimensional fuzzy valued functions, then the granular distance is Dgr(f (t),h(t))= sup λ max αf ,αh ‖fgr(t,λ,αf )−hgr(t,λ,αh)‖, where t ∈ [a,b]⊂R and λ,αf ,αh ∈ [0,1]. Refer to [8] first-order gr-derivative, and gr-integration for n-dimensional fuzzy valued function. Now, we define second order gr-differentiability for n-dimensional fuzzy valued function. Definition 2.8. Let f : [a,b] → RnF, be the n-dimensional fuzzy valued function. If there exists d2grf (t0) dt2 ∈RnF, such that lim h→0 f ′(t0 +h) f ′(t0) h = d2grf (t0) dt2 = f ′′gr(t0), this limit is taken in the metric space (RnF,D n gr). Then f is said to be second order gr- differentiable at a point t0 ∈ [a,b]. Theorem 2.3. Let f : [a,b]→RnF be a n-dimensional fuzzy valued function, then f is gr-differentiable if and only if its HMF is differentiable with respect to t ∈ [a,b]. Moreover, H ( d2grf (t) dt2 ) = ∂2fgr(t,λ,αf ) ∂t2 . 8 Int. J. Anal. Appl. (2023), 21:76 Proof. Assuming that f is second order gr-differentiable then f is first order gr-differentiable and H ( dgrf (t) dt ) = ∂fgr(t,λ,αf ) ∂t . for t ∈ (a,b). Based on the Definition 2.6 and Definition 2.8, for all �1 > 0, there exits δ1 > 0 such that |h| < δ1 =⇒ Dngr( f ′(t+h) f ′(t) h , d2grf (t) dt2 ) < �1 =⇒ sup λ max αf ‖ f ′gr(t +h,λ,αf )− f ′gr(t,λ,αf ) h − d2grfgr(t,λ,αf ) dt2 ‖ < �1 =⇒ ‖ f ′gr(t +h,λ,αf )− f ′gr(t,λ,αf ) h − d2grfgr(t,λ,αf ) dt2 ‖ < �1 =⇒ lim h→0 f ′gr(t +h,λ,αf )− f ′gr(t,λ,αf ) h = d2grfgr(t,λ,αf ) dt2 =⇒ ∂2fgr(t,λ,αf ) ∂t2 = H ( d2grf (t) dt2 ) . � Proposition 2.6. Let f : [a,b] → RnF be a n-dimensional fuzzy valued function defined by f (t) = (f1(t), f2(t), · · · , fn(t)) for all x ∈ [a,b] and fi(t) ∈ RF, with [fi(t)]λ = [ fλil (t), f λ ir (t) ] , i = 1,2, · · · ,n. The n-dimensional fuzzy valued function f is gr-differentiable twice on [a,b] if and only if (fλil ) ′(t) and (fλir ) ′(t) are differentiable on [a,b], for all i =1,2, · · · ,n. Proof. Since fgr(t,λ,αf )= ( f1gr(t,λ,α1), f2gr(t,λ,α2), · · · , fngr(t,λ,αn) ) , then fgr(t,λ,αf )= ( (fλ1l(t)+(f λ 1r (t)− fλ1l(t))α1),(f λ 2l (t)+(fλ2r(t)− f λ 2l (t))α2), · · · , (fλnl (t)+(f λ nr (t)− fλnl (t))αn) ) , where λ, αf , (α1,α2, · · · ,αn)∈ [0,1]. From Definition 2.8 and Theorem 2.3, we have Suppose that f (t) is a gr-differentiable twice on [a,b] ⇐⇒ ∂2fgr(t,λ,αf ) ∂t2 = ( ((fλ1l) ′′(t)+((fλ1r) ′′(t)− (fλ1l) ′(t))α1), ((fλ2l) ′′(t)+((fλ2r) ′′(t)− (fλ2l) ′′(t))α2), · · · ,((fλnl ) ′′(t)+((fλnr) ′′(t)− (fλnl ) ′′(t))αn) ) ⇐⇒ (fλil ) ′(t) and (fλir ) ′(t), are differentiable on [a,b] for i =1,2, · · · ,n. � Definition 2.9. If a matrix A = [aij]n×m, for all aij ∈ RF, i = 1,2, · · · ,n and j = 1,2, · · · ,m. Then that matrix A is called fuzzy matrix. Definition 2.10. If A = [aij]n×m is a fuzzy matrix, then the HMF of A is defined by H(A) = [H(aij)]n×m , [(aij)gr(λ,αij)]n×m, where λ,αij ∈ [0,1], i =1,2, · · · ,n and j =1,2, · · · ,m. Int. J. Anal. Appl. (2023), 21:76 9 3. An algorithm for the solution of system of second order linear fuzzy initial value problems under (SSLFDE) gr-differentiability Consider a SSLFDEs, Z′′gr(t)= A⊗Z(t)⊕F(t), withZ(t0)= Z0. (3.1) The matrix form of (3.1) is, [ y ′′gr(t) z′′gr(t) ] = [ a b c d ] ⊗ [ y(t) z(t) ] ⊕ [ f (t) g(t) ] , (3.2) subject to, [ y(t0) z(t0) ] = [ y0 z0 ] and [ y ′(t0) z′(t0) ] = [ y ′0 z′0 ] . (3.3) The following algorithm describes the procedure to compute λ-cut solution of SSLFDEs (3.1) if it exists. Step 1 : Applying HMF on both sides of (3.2) and (3.3), we get[ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ agr(λ,αa) bgr(λ,αb) cgr(λ,αc) dgr(λ,αd) ][ ygr(t,λ,αy) zgr(t,λ,αz) ] + [ fgr(t,λ,αf ) ggr(t,λ,αg) ] , (3.4) with, [ ygr(t0) zgr(t0) ] = [ y0gr(λ,αy0) z0gr(λ,αz0) ] and [ y ′gr(t0) z′gr(t0) ] = [ y ′0gr(λ,αy ′0 ) z′0gr(λ,αz′0 ) ] , (3.5) where λ, αz,αf , αg, αa, αb,αc, αd, αy0, αz0,αy ′0, αz′0 ∈ [0,1]. Here, (3.4) and (3.5) taken as a ordinary second order system of differential equations. Step 2 : Solving (3.4) and (3.5), we get ygr(t,λ,αy) and zgr(t,λ,αz). (3.6) Step 3 : Applying inverse HMF on both sides of (3.6), we get [y(t)]λ = [ inf λ≤α≤1 min αy ygr(t,α,αy), sup λ≤α≤1 max αy ygr(t,α,αy)], (3.7) [z(t)]λ = [ inf λ≤α≤1 min αz zgr(t,α,αz), sup λ≤α≤1 max αz zgr(t,α,αz)], (3.8) which is the required λ-cut solution of SSLFDEs (3.1). 10 Int. J. Anal. Appl. (2023), 21:76 4. Mechanical applications In this section, we describe mechanical applications [13] of which the uncertain information taken as fuzzy sets. Example 4.1. (Automobile with two axles) Now we have an automobile with two axles and distinct front and back suspension systems, we can examine a more realistic model. The suspension system of such a vehicle is seen in Figure 1 . We suppose that the car’s body behaves similarly to a solid bar with the dimensions of mass M and length l = l1 + l2. Its centre of mass c, which is located at a distance l1 from the front of the vehicle, has a moment of inertia I around it. The vehicle features suspension springs with Hooke’s constants s1 and s2 for the front and back, respectively. Let y(t) represent the car’s vertical displacement from equilibrium while it is moving, and let z(t) represent its angular displacement (in radians) from the horizontal. The equations may then be derived using Newton’s laws of motion for linear and angular acceleration as follows: M �y ′′gr(t)=−(s1 + s2)�y(t)⊕ (s1l1 − s2l2)�z(t), I �z′′gr(t)= (s1l1 − s2l2)�y(t) (s1l 2 1 + s2l 2 2)�z(t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. Suppose that M = 75lb.s2/f t, l1 = 7f t, l2 = 3f t,s1 = s2 = 2000lb/f t, I = 1000f t.lb.s2 and the λ-level sets of fuzzy initial values are [y0]λ = [z0]λ = [3+ λ,5− λ], [y ′0] λ = [5+ λ,7− λ], [z′0] λ = [6+λ,8−λ]. Figure 1. two axles car. Then the matrix equation is, [ y ′′gr(t) z′′gr(t) ] = [ −53.33 106.67 8 −116 ] � [ y(t) z(t) ] , (4.1) , subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] . (4.2) Int. J. Anal. Appl. (2023), 21:76 11 Taking HMF on both sides of (4.1) and (4.2), we have[ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −53.33 106.67 8 −116 ][ ygr(t,λ,αy) zgr(t,λ,αz) ] , (4.3) subject to, [ ygr(0) zgr(0) ] = [ y0gr(λ,α1) z0gr(λ,α1) ] and [ y ′gr(0) z′gr(0) ] = [ y ′0gr(λ,α2) z′0gr(λ,α3) ] , (4.4) where y0gr(λ,α1)= z0gr(λ,α1)= [3+λ+2(1−λ)α1],y ′0gr(λ,α2)= [5+λ+2(1−λ)α2],z ′ 0gr (λ,α3)= [6+λ+2(1−λ)α3], where λ, α1, α2, α3 ∈ [0,1]. The solution for second order system of equations (4.3) and (4.4) are ygr(t,λ,α1,α2,α3) and zgr(t,λ,α1,α2,α3). (4.5) Applying inverse HMF on (4.5), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2,α3 ygr(t,α,α1,α2,α3), sup λ≤α≤1 max α1,α2,α3 ygr(t,α,α1,α2,α3)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2,α3 zgr(t,α,α1,α2,α3), sup λ≤α≤1 max α1,α2,α3 zgr(t,α,α1,α2,α3)]. The λ-level sets solution is enumerated using MATLAB and is illustrated in Figure 2 (a) λ-level sets of y(t). (b) λ-level sets of z(t). Figure 2. The black curve gives the solution at λ =1 for the system (4.1) and (4.2). Example 4.2. (One springs-two railway cars system) Figure 3 represents one spring supporting two railway cars of masses M1 and M2 respectively system to one other. If all two of the two cars rightward displacements from their respective equilibrium positions are positive, then the spring is 12 Int. J. Anal. Appl. (2023), 21:76 extended byy(t). The motion equations for the two cars are generated as follows: M1 �y ′′gr(t)=−s �y(t)⊕ s �z(t), M2 �z′′gr(t)= s �y(t) s �z(t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. Figure 3. One springs-two cars systems. Suppose that M1 =1lb.s2/f t,M2 =1lb.s2/f t, and the λ-level sets of spring constant and fuzzy initial values are [s]λ = [1+λ,3−λ], [y0]λ = [z0]λ = [λ,2−λ], [y ′0] λ = [1+λ,3−λ], [z′0] λ = [λ,2−λ]. Then the matrix equation is,[ 1 0 0 3 ] � [ y ′′gr(t) z′′gr(t) ] = [ −s s s −s ] ⊗ [ y(t) z(t) ] , (4.6) subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] . (4.7) Taking HMF on both sides of (4.6) and (4.7), we have[ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −sgr(λ,α1) sgr(λ,α1) sgr(λ,α1) −sgr(λ,α1) ][ ygr(t,λ,αy) zgr(t,λ,αz) ] , (4.8) subject to [ ygr(0) zgr(0) ] = [ y0gr(λ,α2) z0gr(λ,α2) ] and [ y ′gr(0) z′gr(0) ] = [ y ′0gr(λ,α1) z′0gr(λ,α2) ] , (4.9) here sgr(λ,α1) = y ′0gr(λ,α1) = [1+ λ +2(1− λ)α1], y0gr(λ,α2) = z0gr(λ,α2) = z ′ 0gr (λ,α2) = [λ+2(1−λ)α2], where λ, α1α2 ∈ [0,1]. Int. J. Anal. Appl. (2023), 21:76 13 =⇒ [ ∂2ygr(t,λ,α1,α2) ∂t2 ∂2zgr(t,λ,α1,α2) ∂t2 ] = [ −(1+λ+2(1−λ)α1) 1+λ+2(1−λ)α1 1+λ+2(1−λ)α1 −(1+λ+2(1−λ)α1) ] [ ygr(t,λ,α1,α2) zgr(t,λ,α1,α2) ] , (4.10) subject to, [ ygr(0) zgr(0) ] = [ λ+2(1−λ)α1 λ+2(1−λ)α1 ] and [ y ′gr(0) z′gr(0) ] = [ 1+λ+2(1−λ)α2 λ+2(1−λ)α1 ] . (4.11) The solution for second order system of equations (4.10) and (4.11) is ygr(t,λ,α1,α2) and zgr(t,λ,α1,α2). (4.12) Applying inverse HMF on (4.12), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2 ygr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 ygr(t,α,α1,α2)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2 zgr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 zgr(t,α,α1,α2)]. The λ-level sets solution is enumerated using MATLAB and is illustrated in Figure 4 (a) λ-level sets of y(t). (b) λ-level sets of z(t). Figure 4. The black curve gives the solution at λ =1 for the system (4.6) and (4.7). Example 4.3. (Two springs-two mass systems with external fuzzy force) Figure 5 represents two springs supporting two masses to one other. If all the two masses rightward displacements from their respective equilibrium positions are positive, then (i) The first spring is extended by y(t). (ii) The second spring is extended by z(t) y(t). 14 Int. J. Anal. Appl. (2023), 21:76 The motion equations for the two masses are generated as follows: M1 �y ′′gr(t)=−s1 �y(t)⊕ s2 � (z(t) y(t)), M2 �z′′gr(t)=−s2 � (z(t) y(t))+ f (t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. Figure 5. Two springs-two masses systems. The matrix form of system of equations is[ y ′′gr(t) z′′gr(t) ] = [ s1 s2 s3 s4 ] ⊗ [ y(t) z(t) ] ⊕ [ 0 pcos(10t) ] , (4.13) subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] , (4.14) where λ-cut set of coefficients and initial values are s1 =−3, s2 =1, s3 =1, s4 =−1,[y0]λ = [z0]λ = [y ′0] λ = [λ,2−λ], [z′0] λ = [p]λ = [1+λ,3−λ]. Taking HMF on both sides of (4.13) and (4.14), we have [ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −3 1 1 −1 ][ ygr(t,λ,αy) zgr(t,λ,αz) ] + [ 0 pgr(λ,α1)cos(10t) ] , (4.15) subject to, [ ygr(0) zgr(0) ] = [ y0gr z0gr ] and [ y ′gr(t0) z′gr(t0) ] = [ y ′0gr z′0gr ] , (4.16) where pgr(λ,α1)= [1+λ+2(1−λ)α1], y0gr(λ,α2)= z0gr(λ,α2)= y ′0gr(λ,α2)= [λ+2(1−λ)α2], z′0gr(λ,α2)= [1+λ+2(1−λ)α1], where λ, α1α2 ∈ [0,1]. Int. J. Anal. Appl. (2023), 21:76 15 =⇒ [ ∂2ygr(t,λ,α1,α2) ∂t2 ∂2zgr(t,λ,α1,α2) ∂t2 ] = [ −3 1 1 −1 ][ ygr(t,λ,α1,α2) zgr(t,λ,α1,α2) ] + [ 0 [1+λ+2(1−λ)α1]cos(10t) ] , (4.17) with, [ ygr(0) zgr(0) ] = [ λ+2(1−λ)α2 λ+2(1−λ)α2 ] and [ y ′gr(0) z′gr(0) ] = [ λ+2(1−λ)α2 1+λ+2(1−λ)α1 ] . (4.18) The solution for system of equations (4.17) and (4.18) is ygr(t,λ,α1,α2) and zgr(t,λ,α1,α2). (4.19) Applying inverse HMF on (4.19), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2 ygr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 ygr(t,α,α1,α2)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2 zgr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 zgr(t,α,α1,α2)]. The λ-level sets solution is enumerated using MATLAB and is illustrated in Figure 6 (a) λ-level sets of y(t). (b) λ-level sets of z(t). Figure 6. The black curve gives the solution at λ =1 for the system (4.13) and (4.14). Example 4.4. (Three springs-two mass systems with external fuzzy force) Three springs supporting two masses on both sides and one another is depicts in Figure 7. Assume that there is no friction as the masses move and that each spring abides by Hooke’s law. Let f (t) be the fuzzy force applying on mass M1 at time t ≥ 0. If all the two masses rightward displacements (from their individual equilibrium positions) are positive, then 16 Int. J. Anal. Appl. (2023), 21:76 (i) The first spring is extended by y(t). (ii) The second spring is extended by z(t) y(t). (iii) The third spring is compressed by z(t). The motion equations for the two masses are generated as follows: M1 �y ′′gr(t)=−s1 �y(t)⊕ s2 � (z(t) y(t))+ f (t), M2 �z′′gr(t)=−s2 � (z(t) y(t))− s3 �z(t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. Figure 7. Three springs-two masses systems. The matrix form of system of equations is[ y ′′gr(t) z′′gr(t) ] = [ s1 s2 s3 s4 ] ⊗ [ y(t) z(t) ] ⊕ [ pcos(10t) 0 ] , (4.20) subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] , (4.21) where λ-cut set of coefficients and initial values are s1 =−3, s2 =1, s3 =1, s4 =−3, y0 =1, z0 =1, [y ′0] λ = [λ,2−λ], [z′0] λ = [p]λ = [1+λ,3−λ]. Taking HMF on both sides of (4.20) and (4.21), we have [ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −3 1 1 −3 ][ ygr(t,λ,αy) zgr(t,λ,αz) ] + [ pgr(λ,α2)cos(10t) 0 ] , (4.22) subject to, [ ygr(0) zgr(0) ] = [ y0gr z0gr ] and [ y ′gr(t0) z′gr(t0) ] = [ y ′0gr z′0gr ] , (4.23) where pgr(λ,α2) = [1 + λ + 2(1 − λ)α2], y ′0gr(λ,α1) = [λ + 2(1 − λ)α1], z ′ 0gr (λ,α2) = [1+λ+2(1−λ)α1], where λ, α1α2 ∈ [0,1]. Int. J. Anal. Appl. (2023), 21:76 17 =⇒ [ ∂2ygr(t,λ,α1,α2) ∂t2 ∂2zgr(t,λ,α1,α2) ∂t2 ] = [ −3 1 1 −3 ][ ygr(t,λ,α1,α2) zgr(t,λ,α1,α2) ] + [ [1+λ+2(1−λ)α2]cos(10t) 0 ] , (4.24) with [ ygr(0) zgr(0) ] = [ 1 1 ] and [ y ′gr(0) z′gr(0) ] = [ λ+2(1−λ)α1 1+λ+2(1−λ)α2 ] . (4.25) The solution for system of equations (4.24) and (4.25) is ygr(t,λ,α1,α2) and zgr(t,λ,α1,α2). (4.26) Applying inverse HMF on (4.26), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2 ygr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 ygr(t,α,α1,α2)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2 zgr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 zgr(t,α,α1,α2)]. The λ-level sets solution is enumerated using MATLAB and is illustrated in Figure 8 (a) λ-level sets of y(t). (b) λ-level sets of z(t). Figure 8. The black curve gives the solution at λ =1 for the system (4.20) and (4.21). 5. Conclusions This paper mainly deals with determining solutions of SSLFDEs and applications to some me- chanical problems. The granular differentiability is extended to n-dimensional fuzzy valued functions. The SSLFDEs with fuzzy initial conditions are investigated under gr-differentiability. An algorithm is developed to determine the solutions of SSLFDEs with fuzzy initial conditions. Some mechanical 18 Int. J. Anal. Appl. (2023), 21:76 problems as automobiles with two axles, railway cars systems, and mass-spring systems with fuzzy initial conditions are demonstrated for the effective implementation of the algorithm. In the future, this work will be extended for higher-order SFDEs with fuzzy initial and boundary conditions. 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Penney, Differential Equations and Boundary Value Problems: Computing and Modeling, Fifth ed., Pearson Education, Boston, 2000. https://doi.org/10.1016/j.fss.2021.06.003 https://doi.org/10.1016/j.ins.2011.06.007 https://doi.org/10.1016/j.ins.2011.06.007 https://doi.org/10.48550/arXiv.0910.4307 https://doi.org/10.1142/S1793005721500010 https://doi.org/10.1142/S1793005721500010 https://doi.org/10.1109/TFUZZ.2017.2659731 https://doi.org/10.22436/jmcs.018.02.07 https://doi.org/10.22436/jmcs.018.02.07 https://doi.org/10.28924/2291-8639-21-2023-4 https://doi.org/10.1007/s00500-020-05055-8 https://doi.org/10.1016/j.fss.2021.01.003 https://doi.org/10.1007/s41066-017-0054-5 https://doi.org/10.1007/s41066-021-00293-z 1. Introduction 2. Preliminaries 3. An algorithm for the solution of system of second order linear fuzzy initial value problems under (SSLFDE) gr-differentiability 4. Mechanical applications 5. Conclusions References