Systems of First-Order Linear Fuzzy Initial Value Problems and Their Applications

Nagalakshmi Soma; Suresh Kumar Grande; Ravi P. Agarwal

doi:10.5391/IJFIS.2024.24.2.171

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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 171-180

Published online June 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.2.171

Systems of First-Order Linear Fuzzy Initial Value Problems and Their Applications

Nagalakshmi Soma¹, Suresh Kumar Grande², and Ravi P. Agarwal³

¹Basic Engineering Department, MIC College of Technology, Kanchikacherla, India
²Department of Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, India
³Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX, USA.

Correspondence to :
Suresh Kumar Grande (drgsk006@kluniversity.in)

Received: December 6, 2022; Revised: November 6, 2023; Accepted: February 28, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

This study primarily addresses solutions to a system of first-order linear fuzzy initial value problems in the context of granular differentiability, and explores the real-life applications of such systems. We recall the concepts of the horizontal membership function, granular metrics, limits, continuity, differentiability, and integrability for fuzzy functions with n-dimensional fuzzy numbers. We then present a fundamental theorem that establishes the existence and uniqueness of solutions for both homogeneous and nonhomogeneous systems of first-order linear fuzzy initial value problems. In addition, we describe an algorithm for solving nonhomogeneous systems under granular differentiability. Finally, we provide real-life applications - including models for lidocaine and irregular heartbeats, Richardson’s arms race, and radioactive decay phenomena in a fuzzy environment - to demonstrate the practical utility of the proposed algorithm.

Keywords: Horizontal membership function, n-dimensional granular metric, n-dimensional granular derivative, System of first-order linear fuzzy initial value problems

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Fuzzy sets and arithmetic play important roles in mathematical models that include uncertain or vague variables. Accordingly, fuzzy differential equations (FDEs) and their applications have recently attracted considerable attention. A system of fuzzy differential equations (SFDE) can be derived the various behaviors of many dynamic systems that include uncertainty. Systems of first-order linear fuzzy initial value problems appear in many practical situations such as growth and decay models, bioinformatics, and economics. Fard and Ghal [1] introduced an iterative technique for solving SFDEs with fuzzy constant coefficients using the H-differentiability concept. Mondal et al. [2] analyzed adaptive schemes to study SFDEs using an arm race model. Agarwal et al. [3] conducted a survey of fractional FDEs. Barazandeh and Ghazanfari [4] obtained numerical solutions to SFDEs using a variation iteration technique. Keshavarz et al. [5] proposed an analytical solution for SFDEs with gH-differentiability. Boukezzoula et al. [6] proposed a technique for solving SFDEs using variables as fuzzy intervals. In all aforementioned studies, authors used H-, SGH-, and gH-derivatives for fuzzy functions. However, these derivatives present the following drawbacks: (i) nonexistence of derivatives, (ii) doubling property, (iii) multiplicity of solutions, (iv) monotonicity of uncertainty, and (v) unnatural behavior in modeling (UBM).

Mazandarani et al. [7] established granular differentiability (gr-differentiability) using the horizontal membership function (HMF) to overcome the aforementioned challenges. Subsequently, Mazandarani and Xiu [8] reviewed FDEs. Najariyan and his colleagues [9,10] obtained solutions for singular SFDEs under gr-differentiability. Yang et al. [11] investigated solutions for a linear second-order fuzzy boundary value problem under gr-differentiability. Nagalakshmi et al. [12] introduced a system of first-order linear two-point fuzzy boundary value problems under gr-differentiability. Zhang et al. [13] established optimality conditions for fuzzy optimization problems under granular convexity. Motivated by these efforts, we extended the granular differentiability concept to establish the existence and uniqueness of the solution to a fully fuzzy-valued first-order system of differential equations. We present an algorithm designed to solve this problem, as well as some real-life applications that illustrate its practical use.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

This section presents valuable definitions and notations used to establish the main results.

A nonempty fuzzy subset p of R with membership function p : R → [0, 1] is said to be a fuzzy number if it is semicontinuous, fuzzy convex, normal, and compactly supported on R, where p(y) denotes the membership degree of y for each y ∈ R.

Let R_F denote the space of fuzzy numbers in R. The β-level sets of p are defined by [p]β={y∈R:p(y)≥β}=[plβ,prβ] for 0 < β ≤ 1 and [p]⁰ = cl{y ∈ R : p(y) > 0}.

Fundamental definitions and results related to fuzzy numbers, including standard arithmetic operations and β-levels, are found in [4]. For notations, definitions, and primary results related to the HMF, gr-differentiation, and gr-integration of fuzzy functions, refer to [7]. Motivated by [7], we extended gr-differentiation and gr-integration to n-dimensional fuzzy functions.

Definition 1

Let RFn=RF×RF×RF×⋯×RF︸n times be the space of n^th-order FN vectors. Then, addition and scalar multiplication can be defined component-wise as follows: If u = (u₁, u₂, · · ·, u_n), v=(v1,v2,⋯,vn)∈RFn, then

(i) u+v = (u₁⊕v₁, u₂⊕v₂, · · ·, u_n⊕v_n), where u_i, vi∈RFn, i = 1, 2, · · · n.
(ii) ku = (ku₁, ku₂, · · ·, ku_n), where ui∈RFn, i = 1, · · · n and k ∈ R_F, is a fuzzy scalar.

Definition 2

If u=(u1,u2,⋯,un)∈RFn, as u_i ∈ R_F, i = 1, 2,· · ·, n. Then, the HMF for u∈RFn is defined by u_gr(β,α_u) = (u_1gr(β,α₁), u_2gr(β,α₂), · · ·, u_ngr(β,α_n)), where β, α₁, · · ·, α_n ∈ [0, 1] and α₁, · · ·, α_n are called relative distance measure (RDM) variables.

Definition 3

If g:[b,c]→RFn, is a fuzzy function, it is called an n-dimensional fuzzy function in [b, c].

Definition 4 [12]

If g:[b,c]→RFn is an n-dimensional fuzzy function that includes mn ∈ N distinct fuzzy numbers, then the HMF of g is denoted by H(g(y)) ^≜g_gr(y, β,α_g) and interpreted as ggr:[b,c]×[0,1]×[0,1]×⋯×[0,1]︸mn times→Rn, wherein α_g encompasses mn RDM variables corresponding to mn FNs in the fuzzy function g.

Definition 5 [12]

Let p and q be two-dimensional FNs. Then, H(p) = H(q) for all α_p = α_q ∈ [0, 1] if and only if p=q.

Definition 6 [12]

Let p, q∈RFn. The function Dgrn:RFn×RFn→R+∪{0}, defined by

Dgrn(p,q)=supβ maxαp,αq‖pgr(β,αp)-qgr(β,αq)‖,

is called the n-dimensional granular distance between two n-dimensional FNs p and q, where ||.|| represents the Euclidean norm in Rⁿ.

Proposition 1

Function Dgrn is a metric of space RFn.

Proof

Suppose that RFn is a nonempty set and Dgrn:RFn×RFn→R+∪{0} is a real-valued function.

(i) Consider

Dgrn(p,q)=supβ maxαp,αq‖pgr(β,αp)-qgr(β,αq)‖>0.
(ii) Consider

Dgrn(p,q)=0⇔supβ maxαp,αq‖pgr(β,αp)-qgr(β,αp)‖=0⇔‖pgr-qgr‖=0⇔pgr=qgr⇔p=q.
(iii) Consider

Dgrn(p,q)=supβ maxαp,αq‖pgr(β,αp)-qgr(β,αq)‖=supβ maxαp,αq‖qgr(β,αq)-pgr(β,αp)‖=Dgrn(q,p).
(iv) Consider

Dgrn(p,r)=supβ maxαp,αr‖pgr(β,αp)-rgr(β,αr)‖=supβ maxαp,αq,αr‖pgr(β,αp)-qgr(β,αq)+qgr(β,αq)-rgr(β,αr)‖≤supβ maxαp,αq‖pgr(β,αp)gr-qgr(β,αq)‖+supβ maxαq,αr‖qgr(β,αq)-rgr(β,αr)‖=Dgrn(p,q)+Dgrn(q,r).

From (i) to (iv), (RFn,Dgrn) is a metric space (MS).

Definition 7

Let g:[b,c]→RFn be an n-dimensional fuzzy function. The limit of g(y) as y → p is q∈RFn, and is subject to the following conditions:

(i) If p ∈ (b, c), for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as limy→p g(y)=q.
(ii) If p = b, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as limy→b+ g(y)=q.
(iii) If p = c, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as limy→c- g(y)=q.

Definition 8

Let g:[b,c]→RFn be an n-dimensional fuzzy function. Function g(y) is said to be continuous at y = p if g(p)∈RFn, which is subject to the following conditions:

(i) If p ∈ (b, c), for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as limy→p g(y)=g(p).
(ii) If p = b, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as limy→b+ g(y)=g(b).
(iii) If p = c, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as limy→c- g(y)=g(c).

Remark 1

If g,h:[b,c]→RFn, are n-dimensional fuzzy functions, then the granular distance is

Dgrn(g(y),h(y))=supβ maxαg,αh‖ggr(y,β,αg)-hgr(y,β,αh)‖,

where y ∈ [b, c] ⊂ R and β,α_g, α_h ∈ [0, 1].

Next, we define first-order gr-differentiability for an n-dimensional fuzzy function.

Definition 9 [12]

Let g:[b,c]→RFn be an n-dimensional fuzzy function. If there exists dgrg(y0)dy∈RF such that

limh→0g(y0+h)⊖grg(y0)h=dgrg(y0)dy=ggr′(y0),

this limit is taken in the metric space (RFn,Dgrn), and g is considered first-order gr-differentiable at a point y₀ ∈ [b, c].

Theorem 1

Let g : [b, c] → R_F be an n-dimensional fuzzy function. Then, g is gr-differentiable if and only if its HMF is differentiable with respect to y ∈ [b, c] and

H (dgrg(y)dy)=∂ggr(y,β,αf)∂y.

Proof

Suppose that g is gr-differentiable at y ∈ (b, c). From Definition 9, for each ε₁> 0, there exits δ₁> 0 such that |h| < δ₁

⇒Dngr(g(y+h)⊖grg(y)h,dgrg(y)dy)<ɛ1⇒supβ maxαg‖ggr(y+h,β,αg)-ggr(y,β,αg)h-dgrggr(y,β,αg)dy‖<ɛ1⇒‖ggr(y+h,β,αg)-ggr(y,β,αg)h-dgrggr(y,β,αg)dy‖<ɛ1⇒limh→0ggr(y+h,β,αg)-ggr(y,β,αg)h=dgrggr(y,β,αg)dy⇒∂ggr(y,β,αg)∂y=H (dgrg(y)dy).

Definition 10 [12]

Suppose that g:[b,c]→RFn is continuous and HMF H(g(y)) ^≜g_gr(y, β,α_g) is integrable on [b, c]. If there exists a m such that H(m)=∫bcH(g(y))dy, then m is called the gr-integral of g on [b, c] and m=∫bcg(y)dy.

Proposition 2 [12]

Assume that F:[b,c]→RFn is gr-differentiable and g(y)=dgrF(y)dy is continuous on [b, c]. Then, ∫bcg(y)dy=F(c)⊖grF(b).

Definition 11

If a matrix A = [a_ij ]_n×m, for all a_ij ∈ R_F, i = 1, 2, · · ·, n and j = 1, 2, · · ·, m. Thus, matrix A is considered a fuzzy matrix.

Definition 12

If A = [a_ij ]_n×m is a fuzzy matrix, then the HMF of A is defined as H(A) = [H(a_ij )]_n×m^≜ [(a_ij)_gr(β, α_ij )]_n×m = A_gr(β,α_A), where β, α_ij ∈ [0, 1], i = 1, 2, · · ·, n and j = 1, 2, · · ·, m.

3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Consider the following nonhomogeneous system of first-order linear fuzzy initial value problems:

(1)Zgr′(x)=AZ(x)+F(x),Z(x0)=Z0.

The corresponding homogeneous fuzzy system is expressed as

(2)Zgr′(x)=AZ(x),Z(x0)=Z0.

Here, the product between the fuzzy matrix A = [a_ij ] and column matrix Z(x) = [Z₁(x), Z₂(x), . . . , Z_n(x)]^T is defined as

AZ(x)=[∑j=1na1jZj(x),∑j=1na2jZj(x),…,∑j=1nanjZj(x)]T.

We now prove the existence and uniqueness theorem for the solution of a homogeneous system of first-order linear fuzzy initial value problems (2) under gr-differentiability.

Theorem 2

Let A be an n^th-order square matrix of FNs. Then, for a given Z0∈RFn, the homogeneous fuzzy system in Eq. (2) has a unique solution.

Proof

Taking HMF on both sides of Eq. (2), we obtain

(3)∂Zgr(x,β,αZ)∂x=Agr(β,αA)Zgr(x,β,αZ),

(4)Zgr(x0,β,αZ)=Z0gr(β,αZ0).

We can now prove that Eqs. (3) and (4) have unique solutions. Clearly, the homogeneous fully fuzzy system in Eq. (2) has a unique solution.

Let Z_gr(x, β, α_Z) = exp(A_gr(β,α_A)(x–x₀))Z_{0_gr}(β,α_Z₀),

∂Zgr(x,β,αZ)∂x=∂∂x(exp(Agr(β,αA) (x-x0))Z0gr(β,αZ0))=Agr(β,αA) exp(Agr(β,αA) (x-x0))Z0gr(β,αZ0)=Agr(β,αA)Zgr(x,β,αZ),

for all x > x₀ ∈ R. Additionally, Z_gr(x₀, β, α_Z) = IZ_{0_gr}(β, α_Z₀) = Z_{0_gr}(β, α_Z₀). Therefore, Z_gr(x, β, α_Z)=exp(A_gr(β, α_A)(x – x₀))Z_{0_gr}(β, α_Z₀) is the solution to Eq. (3). We have proven that this solution is unique. Let Y_gr(x, β, α_Y ) be another solution to Eq. (3) and U_gr(x, β, α_U) = exp(−A_gr(β,α_A)(x–x₀))Y_gr(x, β, α_Y ). Then, we have

∂Ugr(x,β,αU)∂x=exp(-Agr(β,αA) (x-x0))∂Ygr(x,β,αY)∂x-Agr(β,αA) exp(-Agr(β,αA) (x-x0))Ygr(x,β,αY)=exp(-Agr(β,αA) (x-x0))Agr(β,αA)Ygr(x,β,αY)-Agr(β,αA) exp(-Agr(β,αA) (x-x0))Ygr(x,β,αY)=(exp(-Agr(β,αA) (x-x0))Agr(β,αA)-Agr(β,αA)exp(-Agr(β,αA) (x-x0)))Ygr(x,β,αY)=0.

Therefore, U_gr(x, β, α_U) is a constant and U_gr(x₀, β, α_U) = IY_gr(x₀, β, α_Y) = Y_gr(x₀, β, α_Y) = Z_{0_gr}(β, α_Z₀). Thus, U_gr(x, β, α_U) = Z_{0_gr}(β, α_Z₀) and now the proof completed from Y_gr(x, β, α_Y) = exp(A_gr(β, α_A)(x – x₀))U_gr(x, β, α_U) = exp(A_gr(β, α_A)(x – x₀))Z_{0_gr}(β, α_Z₀) = Z_gr(x, β, α_Z).

We now prove the existence and uniqueness theorem for the solution to a nonhomogeneous system of first-order linear fuzzy initial value problems (1) under gr-differentiability.

Theorem 3

Let A be an nth-order square matrix of FNs and F:[x0,b]→RFn be a continuous n-dimensional fuzzy function. Then, for a given Z0∈RFn, the nonhomogeneous fuzzy system (1) has a unique solution.

Proof

Taking HMF on both sides of Eq. (1), we obtain

(5)∂Zgr(x,β,αZ)∂x=Agr(β,αA)Zgr(x,β,αZ)+Fgr(x,β,αF),

(6)Zgr(x0,β,αZ)=Z0gr(β,αZ0).

We now prove that Eqs. (5) and (6) have a unique solution, It is clear that the nonhomogeneous fully fuzzy system (1) also has a unique solution. Let

Zgr(x,β,αZ)=exp(Agr(β,αA) (x-x0))Z0gr(β,αZ0)+exp(Agr(β,αA)x)×∫x0xexp(-Agr(β,αA)s)Fgr(s,β,αF)ds.

Then,

∂Zgr(x,β,αZ)∂x=Agr(β,αA) exp(Agr(β,αA) (x-x0))Z0gr(β,αZ0)+exp(Agr(β,αA)x)∫x0xexp(-Agr(β,αA)s)Fgr(s)ds+exp(Agr(β,αA)x) exp(-Agr(β,αA)x)Fgr(x,β,αF)=Agr(β,αA) (exp(Agr(β,αA) (x-x0))Z0gr(β,αZ0)+exp(Agr(β,αA)x)×∫x0xexp(-Agr(β,αA)s)Fgr(s,β,αF)ds)+IFgr(x,β,αF)=Agr(β,αA)Zgr(x,β,αZ)+Fgr(x,β,αF),

for all x > x₀ ∈ R and Z_gr(x₀, β, α_Z) = IZ_{0_gr}(β,α_Z₀) = Z_{0_gr}(β,α_Z₀). Therefore, Z_gr(x, β, α_Z) is the solution to Eqs. (5) and (6). We now prove that Z_gr(x, β, α_Z) is a unique solution to the problem. Let U_gr(x, β, α_U) be another solution. Subsequently, Ugr′(x,β,αU)=Agr(β,αA)Ugr(x,β,αU)+Fgr(x,β,αF), where U_gr(x₀, β, α_U) = Z_{0_gr}(β,α_Z₀). Consider that

(7)∂(Z-U)gr(x,β,αZ-U)∂x=∂Zgr(x,β,αZ)∂x-∂Zgr(x,β,αU)∂x=Agr(β,αA) (Zgr(x,β,αZ)-Ugr(x,β,αU)),

and

(8)(Z-U)gr(x0,β,αZ-U)=0.

Eqs. (7) and (8) represent homogeneous fuzzy systems. From Theorem 2, we obtain Z_gr(x, β, α_Z) – U_gr(x, β, α_U) = 0. This implies that Z_gr(x, β, α_Z) = U_gr(x, β, α_U).

3.1 An Algorithm for Solving Systems of First Order Linear Fuzzy Initial Value Problems

Consider the following nonhomogeneous fuzzy system:

(9)Zgr′(x,β,αZ)=AZ(x)+F(x),with Z(x0)=Z0,

where F:[x0,x1]→RFn denotes a continuous n-dimensional fuzzy function. The matrix form of Eq. (9) (for n = 2) is

(10)[ygr′(x,β,αy)zgr′(x,β,αz)]=[abcd] [y(x)z(x)]+[f(x)g(x)],

(11)subject to [y(x0)z(x0)]=[y0z0],

where a, b, c, d, y₀, z₀, f, g ∈ R_F.

If f (x) and g(x) are fuzzy functions of [x₀, x₁], then from Theorem 3, the nonhomogeneous fully fuzzy systems (10) and Eq. (11) have a unique solution. The following algorithm describes the procedure for computing the β-cut solutions to Eqs. (10) and (11).

Step 1: Applying HMF on both sides of Eqs. (10) and (11), we get

(12)[∂ygr(x,β,αy)∂x∂zgr(x,β,αz)∂x]=[agr(β,αa)bgr(β,αb)cgr(β,αc)dgr(β,αd)] [ygr(x,β,αy)zgr(x,β,αz)]+[fgr(x,β,αf)ggr(x,β,αg)](13)subject to [ygr(x0,β,αy0)zgr(x0,β,αz0)]=[y0gr(β,αy0)z0gr(β,αz0)],

where β, α_f, α_g, α_a, α_b,α_c, α_d, α_y₀, α_z₀∈ [0, 1]. Here, Eq. (12) is a system of partial differential equations with a single independent variable x. Therefore, Eqs. (12) and (13) are taken as an ordinary first-order system of differential equations.
Step 2: Solving Eqs. (12) and (13), we get a unique solution as

(14)ygr(x,β,αy) and zgr(x,β,αz).
Step 3: Applying inverse HMF on both sides of Eq. (14), we get

[y(x)]β=[infβ≤α≤1 minαy ygr(x,α,αy),supβ≤α≤1 maxαy ygr(x,α,αy)],[z(x)]β=[infβ≤α≤1 minαz zgr(x,α,αz),supβ≤α≤1 maxαz zgr(x,α,αz)],

which is the required β-cut solution to the linear SFDE (9).

4. Applications

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Example 1 (Lidocaine and irregular heartbeat model with uncertainty). Lidocaine is clinically used to treat irregular heartbeat. Let y(x) and z(x) be the amount of lidocaine in the bloodstream and body tissue, respectively. The mathematical model represents a homogeneous linear SFDE, as in Example 5.3 [4], which was solved under H-differentiability. However, H-differentiability has many drawbacks such as the doubling property, existence of an H-difference, and the UBM phenomenon. To overcome these drawbacks, we instead used granular differentiability. The corresponding fully fuzzy system can be written as

(15)[ygr′(x)zgr′(x)]=[-k1k2k3-k2] [y(x)z(x)],

(16)subject to [y(0)z(0)]=[0z0],

where the β-cut set of fuzzy coefficients and initial values is [k₁]^β = [0.08 + 0.01β, 0.1 – 0.01β], [k₂]^β = [0.028 + 0.01β, 0.048 – 0.01β], [k₃]^β = [0.056 + 0.01β, 0.076 – 0.01β], [z₀]^β = [0.99 + 0.01β, 1.01 – 0.01β], with β ∈ [0, 1].

We now apply the proposed method to solve this problem using granular differentiability. By taking HMF on both sides of Eqs. (15) and (16), we obtain

(17)[∂ygr(x,β,αy)∂x∂zgr(x,β,αz)∂x]=[-k1gr(β,α1)k2gr(β,α2)k3gr(β,α3)-k2gr(β,α2)] [ygr(x,β,αy)zgr(x,β,αz)],

(18)subject to [ygr(x0)zgr(x0)]=[0z0gr(β,α4)],

where the granule of fuzzy coefficients and initial values is k_{1_gr}= 0.08+0.01β+0.02(1–β)α₁, k_{2_gr}= 0.028+0.01β+ 0.02(1–β)α₂, k_{3_gr}= 0.056+0.01β+0.02(1–β)α₃, z_{0_gr}= [0.99 + 0.01β + 0.02(1 – β)α₄], where β, α₁, α₂, α₃, α₄ ∈ [0, 1].

The solution to the system of Eqs. (17) and (18) is

(19)ygr(x,β,α1,α2,α3,α4) and zgr(x,β,α1,α2,α3,α4).

By applying the inverse HMF to Eq. (19), we obtain

[y(x)]β=[infβ≤α≤1 minα1,α2,α3,α4 ygr(x,α,α1,α2,α3,α4),supβ≤α≤1 maxα1,α2,α3,α4 ygr(x,α,α1,α2,α3,α4)],[z(x)]β=[infβ≤α≤1 minα1,α2,α3,α4 zgr(x,α,α1,α2,α3,α4),supβ≤α≤1 maxα1,α2,α3,α4 zgr(x,α,α1,α2,α3,α4)].

Figures 1 and 2, representing the nature of the increase or decrease and uncertain (β-cut solutions), support the length of lidocaine in the bloodstream and body tissue, respectively.

Example 2 (Richardson’s arms race modal with uncertainty). Considering the nonhomogeneous linear SFDE seen in Example 6 [2], the authors used the concept of gH-differentiability. However, this concept has drawbacks such as the doubling property, multiplicity, and UBM phenomenon of solutions, prompting us to instead use gr-differentiability. Assume that the two nations are engaged in an arms race, with countries I and P spending y(x) and z(x) per year on armaments, respectively. Under the initial conditions of y(0) = y₀ and z(0) = z₀, the matrix form of the model is

(20)[ygr′(x)zgr′(x)]=[-323-4] [y(x)z(x)]+[12],

(21)subject to [y(0)z(0)]=[y0z0],

where the β-cut set of initial values is [y₀]^β = [z₀]^β = [70 + 30β, 130 – 30β] and β ∈ [0, 1].

By taking the HMF on both sides of Eqs. (20) and (21), we obtain

(22)[∂ygr(x,β,αy)∂x∂zgr(x,β,αz)∂x]=[-323-4] [ygr(x,β,αy)zgr(x,β,αz)]+[12],

(23)subject to [ygr(x0)zgr(x0)]=[y0gr(β,α1)z0gr(β,α1)],

where the initial values of the fuzzy granules are y_{0_gr}(β,α₁) = z_{0_gr}(β, α₁)=[70+ 30β + 60(1 – β)α₁] and β, α₁∈[0, 1].

The solution to the system of Eqs. (22) and (23) is

(24)ygr(x,β,α1) and zgr(x,β,α1).

By applying the inverse HMF to Eq. (24), we obtain

[y(x)]β=[infβ≤α≤1 minα1 ygr(x,α,α1),supβ≤α≤1 maxα1 ygr(x,α,α1)],[z(x)]β=[infβ≤α≤1 minα1 zgr(x,α,α1),supβ≤α≤1 maxα1 zgr(x,α,α1)].

The β-cut solutions to Eqs. (20) and (21) are shown in Figures 3 and 4. It is clear that the amount of money spent on armaments by countries I and P per year decreases with the support length of uncertainty.

Example 3 (Radioactive decay phenomena with uncertainty). The accelerator beam is directed towards a target made from a stable nucleus (type 1), and radioactive species with half-lives f(x) are generated. The stable y(x) nuclei decay into unstable z(x) nuclei at a rate of k₁. If a nucleus has a decay constant of k₂, it is considered type 2. f(x) and y(x) are predicted to have a degree of uncertainty due to instrumentation failure. Although the theory of radioactive decay can be used to estimate k₁ and k₂, this method cannot provide precise solutions. Four triangular fuzzy numbers are generated to express the degree of uncertainty associated with these parameters. The mathematical model represents the nonhomogeneous linear SFDE shown in Example 5.3 [1]. Whereas the authors of [1] used the concept of H-differentiability, we instead used granular differentiability. The matrix form of the model is as follows:.

(25)[ygr′(x)zgr′(x)]=[-k10k1-k2] [y(x)z(x)]+[f(x)0],

(26)subject to [y(0)z(0)]=[y00],

where the β-cut set of fuzzy coefficients and initial values is [k₁]^β = [0.2 + 0.1β, 0.4 – 0.1β], [k₂]^β = [0.02 + 0.01β, 0.04 – 0.01β], [f(x)]^β = [4.9 + 0.1β, 5.1 – 0.1β], [y₀]^β = [995 + 5β, 1005 – 5β], where β ∈ [0, 1].

Taking the HMF on both sides of Eqs. (25) and (26), we obtain

(27)[∂ygr(x,β,αy)∂x∂zgr(x,β,αz)∂x]=[-k1gr(β,α1)0k1gr(β,α1)-k2gr(β,α2)] [ygr(x,β,αy)zgr(x,β,αz)]+[fgr(x,β,α3)0],

(28)subject to [ygr(x0)zgr(x0)]=[y0gr(β,α4)0],

where the granule of the fuzzy coefficients and initial values is k_{1_gr}(β,α₁) = 0.2 + 0.1β + 0.2(1 – β)α₁, k_{2_gr}(β,α₂) = 0.02 + 0.01β + 0.02(1 – β)α₂, f_gr(x, β, α₃) = 4.9 + 0.1β + 0.2(1–β)α₃, y_{0_gr}(β,α₄) = 995+5β +0.2(1–β)α₄, where β, α₁, α₂, α₃, α₄ ∈ [0, 1].

The solution to the system of Eqs. (27) and (28) is

(29)ygr(x,β,α1,α2,α3,α4) and zgr(x,β,α1,α2,α3,α4).

Applying the inverse HMF to Eq. (29), we obtain

The β-cut solutions shown in Figures 5 and 6 demonstrate that the nuclei y(x) decrease with increasing support length, whereas the nuclei z(x) increase.

5. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

This study primarily addresses the existence and uniqueness of solutions to fully fuzzy linear systems with fuzzy initial conditions under gr-differentiability. We demonstrated the advantage of gr-differentiability over existing methods. Fundamental theorems provide sufficient conditions for the existence and uniqueness of solutions to both homogeneous and nonhomogeneous fully fuzzy systems. The algorithm represents a systematic procedure for solving these systems, and its performance is demonstrated through applications. In the future, we will extend this methodology to higher-order systems of fuzzy initial and boundary value problems.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

No potential conflict of interest relevant to this article was reported.

Figures

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Fig. 1.

The black curve represents y(x) at β = 1in Example 1.

Fig. 2.

The black curve represents z(x) at β = 1in Example 1.

Fig. 3.

The black curve represents y(x) at β = 1in Example 2.

Fig. 4.

The black curve represents z(x) at β = 1in Example 2.

Fig. 5.

The black curve represents y(x) at β = 1in Example 3.

Fig. 6.

The black curve represents y(x) at β = 1in Example 3.

References

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Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
Figures
References
Biographies

Fard, OS, and Ghal-Eh, N (2011). Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients. Information Sciences. 181, 4765-4779. https://doi.org/10.1016/j.ins.2011.06.007
Mondal, SP, Khan, NA, Razzaq, OA, Tudu, S, and Roy, TK (2018). Adaptive strategies for system of fuzzy differential equation: application of arms race model. Journal of Mathematics and Computer Science. 18, 192-205. https://doi.org/10.22436/jmcs.018.02.07
Agarwal, RP, Baleanu, D, Nieto, JJ, Torres, DF, and Zhou, Y (2018). A survey on fuzzy fractional differential and optimal control nonlocal evolution equations. Journal of Computational and Applied Mathematics. 339, 3-29. https://doi.org/10.1016/j.cam.2017.09.039
Barazandeh, Y, and Ghazanfari, B (2019). Approximate solution for systems of fuzzy differential equations by variational iteration method. Punjab University Journal of Mathematics. 51, 13-33.
Keshavarz, M, Allahviranloo, T, Abbasbandy, S, and Modarressi, MH (2021). A study of fuzzy methods for solving system of fuzzy differential equations. New Mathematics and Natural Computation. 17, 1-27. https://doi.org/10.1142/S1793005721500010
Boukezzoula, R, Jaulin, L, and Coquin, D (2022). A new methodology for solving fuzzy systems of equations: thick fuzzy sets based approach. Fuzzy Sets and Systems. 435, 107-128. https://doi.org/10.1016/j.fss.2021.06.003
Mazandarani, M, Pariz, N, and Kamyad, AV (2018). Granular differentiability of fuzzy-number-valued functions. IEEE Transactions on Fuzzy Systems. 26, 310-323. https://doi.org/10.1109/TFUZZ.2017.2659731
Mazandarani, M, and Xiu, L (2021). A review on fuzzy differential equations. IEEE Access. 9, 62195-62211. https://doi.org/10.1109/ACCESS.2021.3074245
Najariyan, M, and Zhao, Y (2021). Granular fuzzy PID controller. Expert Systems with Applications. 167. article no 114182. https://doi.org/10.1016/j.eswa.2020.114182
Najariyan, M, Pariz, N, and Vu, H (2022). Fuzzy linear singular differential equations under granular differentiability concept. Fuzzy Sets and Systems. 429, 169-187. https://doi.org/10.1016/j.fss.2021.01.003
Yang, H, Wang, F, and Gong, Z (2022). Solving the BVP to a class of second-order linear fuzzy differential equations under granular differentiability concept. Journal of Intelligent & Fuzzy Systems. 42, 5483-5499. https://doi.org/10.3233/JIFS-211958
Nagalakshmi, S, Kumar, GS, and Madhavi, B (2023). Solution for a system of first-order linear fuzzy boundary value problems. International Journal of Analysis and Applications. 21. article no 4. https://doi.org/10.28924/2291-8639-21-2023-4
Zhang, J, Chen, X, Li, L, and Ma, X (2022). Optimality conditions for fuzzy optimization problems under granular convexity concept. Fuzzy Sets and Systems. 447, 54-75. https://doi.org/10.1016/j.fss.2022.01.004

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1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion
Conflict of Interest
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Nagalakshmi Soma is an assistant professor at the Basic Engineering Department, MIC College of Technology, India. She majored in application-oriented mathematics for her Master’s degree, and has conducted research related to fuzzy initial and boundary value problems.

Suresh Kumar Grande is an associate professor at the Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, India. He is a reviewer of many international journals and has published more than 50 research articles in peer-reviewed journals. His research focuses on fuzzy sets, fuzzy systems, differential and difference equations, dynamical systems, and mathematical modeling.

Ravi P. Agarwal is currently a professor at the Department of Mathematics, Texas A&M University-Kingsville, USA. He has published 42 research monographs and more than 1,700 publications (with almost 500 mathematicians worldwide) in prestigious national and international mathematics journals. He has served as an editor or associate editor for more than 40 journals, and has published 24 books. His research focuses on nonlinear analysis, differential and difference equations, fixed-point theory, general inequalities, and fuzzy sets and systems.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 171-180

Published online June 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.2.171

Systems of First-Order Linear Fuzzy Initial Value Problems and Their Applications

Nagalakshmi Soma¹, Suresh Kumar Grande², and Ravi P. Agarwal³

Correspondence to:Suresh Kumar Grande (drgsk006@kluniversity.in)

Received: December 6, 2022; Revised: November 6, 2023; Accepted: February 28, 2024

Abstract

Keywords: Horizontal membership function, n-dimensional granular metric, n-dimensional granular derivative, System of first-order linear fuzzy initial value problems

1. Introduction

2. Preliminaries

This section presents valuable definitions and notations used to establish the main results.

Let R_F denote the space of fuzzy numbers in R. The β-level sets of p are defined by ${[p]}^{β} = {y \in R : p (y) \geq β} = [p_{l}^{β}, p_{r}^{β}]$ for 0 < β ≤ 1 and [p]⁰ = cl{y ∈ R : p(y) > 0}.

Definition 1

Let $R_{F}^{n} = \underset{n t i m e s}{\underset{︸}{R_{F} \times R_{F} \times R_{F} \times \dots \times R_{F}}}$ be the space of n^th-order FN vectors. Then, addition and scalar multiplication can be defined component-wise as follows: If u = (u₁, u₂, · · ·, u_n), $v = (v_{1}, v_{2}, \dots, v_{n}) \in R_{F}^{n}$ , then

(i) u+v = (u₁⊕v₁, u₂⊕v₂, · · ·, u_n⊕v_n), where u_i, $v_{i} \in R_{F}^{n}$ , i = 1, 2, · · · n.
(ii) ku = (ku₁, ku₂, · · ·, ku_n), where $u_{i} \in R_{F}^{n}$ , i = 1, · · · n and k ∈ R_F, is a fuzzy scalar.

Definition 2

If $u = (u_{1}, u_{2}, \dots, u_{n}) \in R_{F}^{n}$ , as u_i ∈ R_F, i = 1, 2,· · ·, n. Then, the HMF for $u \in R_{F}^{n}$ is defined by u_gr(β,α_u) = (u_1gr(β,α₁), u_2gr(β,α₂), · · ·, u_ngr(β,α_n)), where β, α₁, · · ·, α_n ∈ [0, 1] and α₁, · · ·, α_n are called relative distance measure (RDM) variables.

Definition 3

If $g : [b, c] \to R_{F}^{n}$ , is a fuzzy function, it is called an n-dimensional fuzzy function in [b, c].

Definition 4 [12]

If $g : [b, c] \to R_{F}^{n}$ is an n-dimensional fuzzy function that includes mn ∈ N distinct fuzzy numbers, then the HMF of g is denoted by H(g(y)) ^≜g_gr(y, β,α_g) and interpreted as $g_{g r} : [b, c] \times [0, 1] \times \underset{m n t i m e s}{\underset{︸}{[0, 1] \times \dots \times [0, 1]}} \to R^{n}$ , wherein α_g encompasses mn RDM variables corresponding to mn FNs in the fuzzy function g.

Definition 5 [12]

Let p and q be two-dimensional FNs. Then, H(p) = H(q) for all α_p = α_q ∈ [0, 1] if and only if p=q.

Definition 6 [12]

Let p, $q \in R_{F}^{n}$ . The function $D_{g r}^{n} : R_{F}^{n} \times R_{F}^{n} \to R^{+} \cup {0}$ , defined by

D_{g r}^{n} (p, q) = sup_{β} max_{α_{p}, α_{q}} ‖ p_{g r} (β, α_{p}) - q_{g r} (β, α_{q}) ‖,

is called the n-dimensional granular distance between two n-dimensional FNs p and q, where ||.|| represents the Euclidean norm in Rⁿ.

Proposition 1

Function $D_{g r}^{n}$ is a metric of space $R_{F}^{n}$ .

Proof

Suppose that $R_{F}^{n}$ is a nonempty set and $D_{g r}^{n} : R_{F}^{n} \times R_{F}^{n} \to R^{+} \cup {0}$ is a real-valued function.

(i) Consider

$D_{g r}^{n} (p, q) = sup_{β} max_{α_{p}, α_{q}} ‖ p_{g r} (β, α_{p}) - q_{g r} (β, α_{q}) ‖ > 0.$
(ii) Consider

$\begin{array}{l} D_{g r}^{n} (p, q) = 0 \\ \Leftrightarrow sup_{β} max_{α_{p}, α_{q}} ‖ p_{g r} (β, α_{p}) - q_{g r} (β, α_{p}) ‖ = 0 \\ \Leftrightarrow ‖ p_{g r} - q_{g r} ‖ = 0 \\ \Leftrightarrow p_{g r} = q_{g r} \\ \Leftrightarrow p = q . \end{array}$
(iii) Consider

$\begin{array}{l} D_{g r}^{n} (p, q) = sup_{β} max_{α_{p}, α_{q}} ‖ p_{g r} (β, α_{p}) - q_{g r} (β, α_{q}) ‖ \\ = sup_{β} max_{α_{p}, α_{q}} ‖ q_{g r} (β, α_{q}) - p_{g r} (β, α_{p}) ‖ \\ = D_{g r}^{n} (q, p) . \end{array}$
(iv) Consider

$\begin{array}{l} D_{g r}^{n} (p, r) = sup_{β} max_{α_{p}, α_{r}} ‖ p_{g r} (β, α_{p}) - r_{g r} (β, α_{r}) ‖ \\ = sup_{β} max_{α_{p}, α_{q}, α_{r}} ‖ p_{g r} (β, α_{p}) - q_{g r} (β, α_{q}) + q_{g r} (β, α_{q}) - r_{g r} (β, α_{r}) ‖ \\ \leq sup_{β} max_{α_{p}, α_{q}} ‖ p_{g r} {(β, α_{p})}_{g r} - q_{g r} (β, α_{q}) ‖ + sup_{β} max_{α q, α_{r}} ‖ q_{g r} (β, α_{q}) - r_{g r} (β, α_{r}) ‖ \\ = D_{g r}^{n} (p, q) + D_{g r}^{n} (q, r) . \end{array}$

From (i) to (iv), ( $R_{F}^{n}, D_{g r}^{n}$ ) is a metric space (MS).

Definition 7

Let $g : [b, c] \to R_{F}^{n}$ be an n-dimensional fuzzy function. The limit of g(y) as y → p is $q \in R_{F}^{n}$ , and is subject to the following conditions:

(i) If p ∈ (b, c), for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as $lim_{y \to p} g (y) = q$ .
(ii) If p = b, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as $lim_{y \to b +} g (y) = q$ .
(iii) If p = c, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as $lim_{y \to c -} g (y) = q$ .

Definition 8

Let $g : [b, c] \to R_{F}^{n}$ be an n-dimensional fuzzy function. Function g(y) is said to be continuous at y = p if $g (p) \in R_{F}^{n}$ , which is subject to the following conditions:

(i) If p ∈ (b, c), for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as $lim_{y \to p} g (y) = g (p)$ .
(ii) If p = b, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as $lim_{y \to b +} g (y) = g (b)$ .
(iii) If p = c, for all ε₁> 0, there exits δ₁> 0 such that . In this case, it is expressed as $lim_{y \to c -} g (y) = g (c)$ .

Remark 1

If $g, h : [b, c] \to R_{F}^{n}$ , are n-dimensional fuzzy functions, then the granular distance is

D_{g r}^{n} (g (y), h (y)) = sup_{β} max_{α_{g}, α_{h}} ‖ g_{g r} (y, β, α_{g}) - h_{g r} (y, β, α_{h}) ‖,

where y ∈ [b, c] ⊂ R and β,α_g, α_h ∈ [0, 1].

Next, we define first-order gr-differentiability for an n-dimensional fuzzy function.

Definition 9 [12]

Let $g : [b, c] \to R_{F}^{n}$ be an n-dimensional fuzzy function. If there exists $\frac{d_{g r} g (y_{0})}{d y} \in R_{F}$ such that

lim_{h \to 0} \frac{g (y_{0} + h) ⊖_{g r} g (y_{0})}{h} = \frac{d_{g r} g (y_{0})}{d y} = g_{g r}^{'} (y_{0}),

this limit is taken in the metric space ( $R_{F}^{n}, D_{g r}^{n}$ ), and g is considered first-order gr-differentiable at a point y₀ ∈ [b, c].

Theorem 1

Let g : [b, c] → R_F be an n-dimensional fuzzy function. Then, g is gr-differentiable if and only if its HMF is differentiable with respect to y ∈ [b, c] and

H (\frac{d_{g r} g (y)}{d y}) = \frac{\partial g_{g r} (y, β, α_{f})}{\partial y} .

Proof

Suppose that g is gr-differentiable at y ∈ (b, c). From Definition 9, for each ε₁> 0, there exits δ₁> 0 such that |h| < δ₁

\begin{array}{l} \Rightarrow {D^{n}}_{g r} (\frac{g (y + h) ⊖_{g r} g (y)}{h}, \frac{d_{g r} g (y)}{d y}) < ɛ_{1} \\ \Rightarrow sup_{β} max_{α_{g}} ‖ \frac{g_{g r} (y + h, β, α_{g}) - g_{g r} (y, β, α_{g})}{h} - \frac{d_{g r} g_{g r} (y, β, α_{g})}{d y} ‖ < ɛ_{1} \\ \Rightarrow ‖ \frac{g_{g r} (y + h, β, α_{g}) - g_{g r} (y, β, α_{g})}{h} - \frac{d_{g r} g_{g r} (y, β, α_{g})}{d y} ‖ < ɛ_{1} \\ \Rightarrow lim_{h \to 0} \frac{g_{g r} (y + h, β, α_{g}) - g_{g r} (y, β, α_{g})}{h} = \frac{d_{g r} g_{g r} (y, β, α_{g})}{d y} \\ \Rightarrow \frac{\partial g_{g r} (y, β, α_{g})}{\partial y} = H (\frac{d_{g r} g (y)}{d y}) . \end{array}

Definition 10 [12]

Suppose that $g : [b, c] \to R_{F}^{n}$ is continuous and HMF H(g(y)) ^≜g_gr(y, β,α_g) is integrable on [b, c]. If there exists a m such that $H (m) = \int_{b}^{c} H (g (y)) d y$ , then m is called the gr-integral of g on [b, c] and $m = \int_{b}^{c} g (y) d y$ .

Proposition 2 [12]

Assume that $F : [b, c] \to R_{F}^{n}$ is gr-differentiable and $g (y) = \frac{d_{g r} F (y)}{d y}$ is continuous on [b, c]. Then, $\int_{b}^{c} g (y) d y = F (c) ⊖_{g r} F (b)$ .

Definition 11

If a matrix A = [a_ij ]_n×m, for all a_ij ∈ R_F, i = 1, 2, · · ·, n and j = 1, 2, · · ·, m. Thus, matrix A is considered a fuzzy matrix.

Definition 12

3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems

Consider the following nonhomogeneous system of first-order linear fuzzy initial value problems:

(1)

Z_{g r}^{'} (x) = A Z (x) + F (x), Z (x_{0}) = Z_{0} .

The corresponding homogeneous fuzzy system is expressed as

(2)

Z_{g r}^{'} (x) = A Z (x), Z (x_{0}) = Z_{0} .

Here, the product between the fuzzy matrix A = [a_ij ] and column matrix Z(x) = [Z₁(x), Z₂(x), . . . , Z_n(x)]^T is defined as

A Z (x) = {[\sum_{j = 1}^{n} a_{1 j} Z_{j} (x), \sum_{j = 1}^{n} a_{2 j} Z_{j} (x), \dots, \sum_{j = 1}^{n} a_{n j} Z_{j} (x)]}^{T} .

We now prove the existence and uniqueness theorem for the solution of a homogeneous system of first-order linear fuzzy initial value problems (2) under gr-differentiability.

Theorem 2

Let A be an n^th-order square matrix of FNs. Then, for a given $Z_{0} \in R_{F}^{n}$ , the homogeneous fuzzy system in Eq. (2) has a unique solution.

Proof

Taking HMF on both sides of Eq. (2), we obtain

(3)

\frac{\partial Z_{g r} (x, β, α_{Z})}{\partial x} = A_{g r} (β, α_{A}) Z_{g r} (x, β, α_{Z}),

(4)

Z_{g r} (x_{0}, β, α_{Z}) = Z_{0_{g r}} (β, α_{Z_{0}}) .

We can now prove that Eqs. (3) and (4) have unique solutions. Clearly, the homogeneous fully fuzzy system in Eq. (2) has a unique solution.

Let Z_gr(x, β, α_Z) = exp(A_gr(β,α_A)(x–x₀))Z_{0_gr}(β,α_Z₀),

\begin{array}{l} \frac{\partial Z_{g r} (x, β, α_{Z})}{\partial x} = \frac{\partial}{\partial x} (exp (A_{g r} (β, α_{A}) (x - x_{0})) Z_{0_{g r}} (β, α_{Z_{0}})) \\ = A_{g r} (β, α_{A}) exp (A_{g r} (β, α_{A}) (x - x_{0})) Z_{0_{g r}} (β, α_{Z_{0}}) \\ = A_{g r} (β, α_{A}) Z_{g r} (x, β, α_{Z}), \end{array}

\begin{array}{l} \frac{\partial U_{g r} (x, β, α_{U})}{\partial x} = exp (- A_{g r} (β, α_{A}) (x - x_{0})) \frac{\partial Y_{g r} (x, β, α_{Y})}{\partial x} - \\ A_{g r} (β, α_{A}) exp (- A_{g r} (β, α_{A}) (x - x_{0})) Y_{g r} (x, β, α_{Y}) \\ = exp (- A_{g r} (β, α_{A}) (x - x_{0})) A_{g r} (β, α_{A}) Y_{g r} (x, β, α_{Y}) - \\ A_{g r} (β, α_{A}) exp (- A_{g r} (β, α_{A}) (x - x_{0})) Y_{g r} (x, β, α_{Y}) \\ = (exp (- A_{g r} (β, α_{A}) (x - x_{0})) A_{g r} (β, α_{A}) - A_{g r} (β, α_{A}) \\ exp (- A_{g r} (β, α_{A}) (x - x_{0}))) Y_{g r} (x, β, α_{Y}) \\ = 0. \end{array}

We now prove the existence and uniqueness theorem for the solution to a nonhomogeneous system of first-order linear fuzzy initial value problems (1) under gr-differentiability.

Theorem 3

Let A be an nth-order square matrix of FNs and $F : [x_{0}, b] \to R_{F}^{n}$ be a continuous n-dimensional fuzzy function. Then, for a given $Z_{0} \in R_{F}^{n}$ , the nonhomogeneous fuzzy system (1) has a unique solution.

Proof

Taking HMF on both sides of Eq. (1), we obtain

(5)

\frac{\partial Z_{g r} (x, β, α_{Z})}{\partial x} = A_{g r} (β, α_{A}) Z_{g r} (x, β, α_{Z}) + F_{g r} (x, β, α_{F}),

(6)

Z_{g r} (x_{0}, β, α_{Z}) = Z_{0_{g r}} (β, α_{Z_{0}}) .

We now prove that Eqs. (5) and (6) have a unique solution, It is clear that the nonhomogeneous fully fuzzy system (1) also has a unique solution. Let

\begin{array}{l} Z_{g r} (x, β, α_{Z}) \\ = exp (A_{g r} (β, α_{A}) (x - x_{0})) Z_{0_{g r}} (β, α_{Z_{0}}) \\ + exp (A_{g r} (β, α_{A}) x) \\ \times \int_{x_{0}}^{x} exp (- A_{g r} (β, α_{A}) s) F_{g r} (s, β, α_{F}) d s . \end{array}

Then,

\begin{array}{l} \frac{\partial Z_{g r} (x, β, α_{Z})}{\partial x} \\ = A_{g r} (β, α_{A}) exp (A_{g r} (β, α_{A}) (x - x_{0})) Z_{0_{g r}} (β, α_{Z_{0}}) \\ + exp (A_{g r} (β, α_{A}) x) \int_{x_{0}}^{x} exp (- A_{g r} (β, α_{A}) s) F_{g r} (s) d s \\ + exp (A_{g r} (β, α_{A}) x) exp (- A_{g r} (β, α_{A}) x) F_{g r} (x, β, α_{F}) \\ = A_{g r} (β, α_{A}) (exp (A_{g r} (β, α_{A}) (x - x_{0})) Z_{0_{g r}} (β, α_{Z_{0}}) \\ + exp (A_{g r} (β, α_{A}) x) \\ \times \int_{x_{0}}^{x} exp (- A_{g r} (β, α_{A}) s) F_{g r} (s, β, α_{F}) d s) \\ + I F_{g r} (x, β, α_{F}) \\ = A_{g r} (β, α_{A}) Z_{g r} (x, β, α_{Z}) + F_{g r} (x, β, α_{F}), \end{array}

for all x > x₀ ∈ R and Z_gr(x₀, β, α_Z) = IZ_{0_gr}(β,α_Z₀) = Z_{0_gr}(β,α_Z₀). Therefore, Z_gr(x, β, α_Z) is the solution to Eqs. (5) and (6). We now prove that Z_gr(x, β, α_Z) is a unique solution to the problem. Let U_gr(x, β, α_U) be another solution. Subsequently, $U_{g r}^{'} (x, β, α_{U}) = A_{g r} (β, α_{A}) U_{g r} (x, β, α_{U}) + F_{g r} (x, β, α_{F})$ , where U_gr(x₀, β, α_U) = Z_{0_gr}(β,α_Z₀). Consider that

(7)

\begin{array}{l} \frac{\partial {(Z - U)}_{g r} (x, β, α_{Z - U})}{\partial x} \\ = \frac{\partial Z_{g r} (x, β, α_{Z})}{\partial x} - \frac{\partial Z_{g r} (x, β, α_{U})}{\partial x} \\ = A_{g r} (β, α_{A}) (Z_{g r} (x, β, α_{Z}) - U_{g r} (x, β, α_{U})), \end{array}

and

(8)

{(Z - U)}_{g r} (x_{0}, β, α_{Z - U}) = 0.

Eqs. (7) and (8) represent homogeneous fuzzy systems. From Theorem 2, we obtain Z_gr(x, β, α_Z) – U_gr(x, β, α_U) = 0. This implies that Z_gr(x, β, α_Z) = U_gr(x, β, α_U).

3.1 An Algorithm for Solving Systems of First Order Linear Fuzzy Initial Value Problems

Consider the following nonhomogeneous fuzzy system:

(9)

Z_{g r}^{'} (x, β, α_{Z}) = A Z (x) + F (x), with Z (x_{0}) = Z_{0},

where $F : [x_{0}, x_{1}] \to R_{F}^{n}$ denotes a continuous n-dimensional fuzzy function. The matrix form of Eq. (9) (for n = 2) is

(10)

[\begin{matrix} y_{g r}^{'} (x, β, α_{y}) \\ z_{g r}^{'} (x, β, α_{z}) \end{matrix}] = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} y (x) \\ z (x) \end{matrix}] + [\begin{matrix} f (x) \\ g (x) \end{matrix}],

(11)

subject to [\begin{matrix} y (x_{0}) \\ z (x_{0}) \end{matrix}] = [\begin{matrix} y_{0} \\ z_{0} \end{matrix}],

where a, b, c, d, y₀, z₀, f, g ∈ R_F.

Step 1: Applying HMF on both sides of Eqs. (10) and (11), we get

(12) $[\begin{matrix} \frac{\partial y_{g r} (x, β, α_{y})}{\partial x} \\ \frac{\partial z_{g r} (x, β, α_{z})}{\partial x} \end{matrix}] = [\begin{matrix} a_{g r} (β, α_{a}) & b_{g r} (β, α_{b}) \\ c_{g r} (β, α_{c}) & d_{g r} (β, α_{d}) \end{matrix}] [\begin{matrix} y_{g r} (x, β, α_{y}) \\ z_{g r} (x, β, α_{z}) \end{matrix}] + [\begin{matrix} f_{g r} (x, β, α_{f}) \\ g_{g r} (x, β, α_{g}) \end{matrix}]$ (13) $subject to [\begin{matrix} y_{g r} (x_{0}, β, α_{y_{0}}) \\ z_{g r} (x_{0}, β, α_{z_{0}}) \end{matrix}] = [\begin{matrix} y_{0_{g r}} (β, α_{y_{0}}) \\ z_{0_{g r}} (β, α_{z_{0}}) \end{matrix}],$

where β, α_f, α_g, α_a, α_b,α_c, α_d, α_y₀, α_z₀∈ [0, 1]. Here, Eq. (12) is a system of partial differential equations with a single independent variable x. Therefore, Eqs. (12) and (13) are taken as an ordinary first-order system of differential equations.
Step 2: Solving Eqs. (12) and (13), we get a unique solution as

(14) $y_{g r} (x, β, α_{y}) and z_{g r} (x, β, α_{z}) .$
Step 3: Applying inverse HMF on both sides of Eq. (14), we get

$\begin{array}{l} {[y (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{y}} y_{g r} (x, α, α_{y}), sup_{β \leq α \leq 1} max_{α_{y}} y_{g r} (x, α, α_{y})], \\ {[z (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{z}} z_{g r} (x, α, α_{z}), sup_{β \leq α \leq 1} max_{α_{z}} z_{g r} (x, α, α_{z})], \end{array}$

which is the required β-cut solution to the linear SFDE (9).

4. Applications

(15)

[\begin{matrix} y_{g r}^{'} (x) \\ z_{g r}^{'} (x) \end{matrix}] = [\begin{matrix} - k_{1} & k_{2} \\ k_{3} & - k_{2} \end{matrix}] [\begin{matrix} y (x) \\ z (x) \end{matrix}],

(16)

subject to [\begin{matrix} y (0) \\ z (0) \end{matrix}] = [\begin{matrix} 0 \\ z_{0} \end{matrix}],

We now apply the proposed method to solve this problem using granular differentiability. By taking HMF on both sides of Eqs. (15) and (16), we obtain

(17)

[\begin{matrix} \frac{\partial y_{g r} (x, β, α_{y})}{\partial x} \\ \frac{\partial z_{g r} (x, β, α_{z})}{\partial x} \end{matrix}] = [\begin{matrix} - k_{1_{g r}} (β, α_{1}) & k_{2_{g r}} (β, α_{2}) \\ k_{3_{g r}} (β, α_{3}) & - k_{2_{g r}} (β, α_{2}) \end{matrix}] [\begin{matrix} y_{g r} (x, β, α_{y}) \\ z_{g r} (x, β, α_{z}) \end{matrix}],

(18)

subject to [\begin{matrix} y_{g r} (x_{0}) \\ z_{g r} (x_{0}) \end{matrix}] = [\begin{matrix} 0 \\ z_{0_{g r}} (β, α_{4}) \end{matrix}],

The solution to the system of Eqs. (17) and (18) is

(19)

y_{g r} (x, β, α_{1}, α_{2}, α_{3}, α_{4}) and z_{g r} (x, β, α_{1}, α_{2}, α_{3}, α_{4}) .

By applying the inverse HMF to Eq. (19), we obtain

\begin{array}{l} {[y (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{1}, α_{2}, α_{3}, α_{4}} y_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4}), sup_{β \leq α \leq 1} max_{α_{1}, α_{2}, α_{3}, α_{4}} y_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4})], \\ {[z (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{1}, α_{2}, α_{3}, α_{4}} z_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4}), sup_{β \leq α \leq 1} max_{α_{1}, α_{2}, α_{3}, α_{4}} z_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4})] . \end{array}

Figures 1 and 2, representing the nature of the increase or decrease and uncertain (β-cut solutions), support the length of lidocaine in the bloodstream and body tissue, respectively.

(20)

[\begin{matrix} y_{g r}^{'} (x) \\ z_{g r}^{'} (x) \end{matrix}] = [\begin{matrix} - 3 & 2 \\ 3 & - 4 \end{matrix}] [\begin{matrix} y (x) \\ z (x) \end{matrix}] + [\begin{matrix} 1 \\ 2 \end{matrix}],

(21)

subject to [\begin{matrix} y (0) \\ z (0) \end{matrix}] = [\begin{matrix} y_{0} \\ z_{0} \end{matrix}],

where the β-cut set of initial values is [y₀]^β = [z₀]^β = [70 + 30β, 130 – 30β] and β ∈ [0, 1].

By taking the HMF on both sides of Eqs. (20) and (21), we obtain

(22)

[\begin{matrix} \frac{\partial y_{g r} (x, β, α_{y})}{\partial x} \\ \frac{\partial z_{g r} (x, β, α_{z})}{\partial x} \end{matrix}] = [\begin{matrix} - 3 & 2 \\ 3 & - 4 \end{matrix}] [\begin{matrix} y_{g r} (x, β, α_{y}) \\ z_{g r} (x, β, α_{z}) \end{matrix}] + [\begin{matrix} 1 \\ 2 \end{matrix}],

(23)

subject to [\begin{matrix} y_{g r} (x_{0}) \\ z_{g r} (x_{0}) \end{matrix}] = [\begin{matrix} y_{0_{g r}} (β, α_{1}) \\ z_{0_{g r}} (β, α_{1}) \end{matrix}],

where the initial values of the fuzzy granules are y_{0_gr}(β,α₁) = z_{0_gr}(β, α₁)=[70+ 30β + 60(1 – β)α₁] and β, α₁∈[0, 1].

The solution to the system of Eqs. (22) and (23) is

(24)

y_{g r} (x, β, α_{1}) and z_{g r} (x, β, α_{1}) .

By applying the inverse HMF to Eq. (24), we obtain

\begin{array}{l} {[y (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{1}} y_{g r} (x, α, α_{1}), sup_{β \leq α \leq 1} max_{α_{1}} y_{g r} (x, α, α_{1})], \\ {[z (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{1}} z_{g r} (x, α, α_{1}), sup_{β \leq α \leq 1} max_{α_{1}} z_{g r} (x, α, α_{1})] . \end{array}

(25)

[\begin{matrix} y_{g r}^{'} (x) \\ z_{g r}^{'} (x) \end{matrix}] = [\begin{matrix} - k_{1} & 0 \\ k_{1} & - k_{2} \end{matrix}] [\begin{matrix} y (x) \\ z (x) \end{matrix}] + [\begin{matrix} f (x) \\ 0 \end{matrix}],

(26)

subject to [\begin{matrix} y (0) \\ z (0) \end{matrix}] = [\begin{matrix} y_{0} \\ 0 \end{matrix}],

Taking the HMF on both sides of Eqs. (25) and (26), we obtain

(27)

[\begin{matrix} \frac{\partial y_{g r} (x, β, α_{y})}{\partial x} \\ \frac{\partial z_{g r} (x, β, α_{z})}{\partial x} \end{matrix}] = [\begin{matrix} - k_{1_{g r}} (β, α_{1}) & 0 \\ k_{1_{g r}} (β, α_{1}) & - k_{2_{g r}} (β, α_{2}) \end{matrix}] [\begin{matrix} y_{g r} (x, β, α_{y}) \\ z_{g r} (x, β, α_{z}) \end{matrix}] + [\begin{matrix} f_{g r} (x, β, α_{3}) \\ 0 \end{matrix}],

(28)

subject to [\begin{matrix} y_{g r} (x_{0}) \\ z_{g r} (x_{0}) \end{matrix}] = [\begin{matrix} y_{0_{g r}} (β, α_{4}) \\ 0 \end{matrix}],

The solution to the system of Eqs. (27) and (28) is

(29)

y_{g r} (x, β, α_{1}, α_{2}, α_{3}, α_{4}) and z_{g r} (x, β, α_{1}, α_{2}, α_{3}, α_{4}) .

Applying the inverse HMF to Eq. (29), we obtain

\begin{array}{l} {[y (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{1}, α_{2}, α_{3}, α_{4}} y_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4}), sup_{β \leq α \leq 1} max_{α_{1}, α_{2}, α_{3}, α_{4}} y_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4})], \\ {[z (x)]}^{β} = [inf_{β \leq α \leq 1} min_{α_{1}, α_{2}, α_{3}, α_{4}} z_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4}), sup_{β \leq α \leq 1} max_{α_{1}, α_{2}, α_{3}, α_{4}} z_{g r} (x, α, α_{1}, α_{2}, α_{3}, α_{4})] . \end{array}

The β-cut solutions shown in Figures 5 and 6 demonstrate that the nuclei y(x) decrease with increasing support length, whereas the nuclei z(x) increase.

5. Conclusion

Abstract
1. Introduction
2. Preliminaries
3. Fundamental Theorems for Systems of First-Order Linear Fuzzy Initial Value Problems
4. Applications
5. Conclusion

Fig 1.

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