Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

V. Padmapriya; M. Kaliyappan

doi:10.5391/IJFIS.2022.22.2.144

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 144-154

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.144

Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

V. Padmapriya^1,² and M. Kaliyappan³

¹Vellore Institute of Technology, Chennai Campus, India
²New Prince Shri Bhavani Arts and Sciences College, Chennai, India
³Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India

Correspondence to :
V. Padmapriya (v.padmapriya2015@vit.ac.in)

Received: August 18, 2021; Revised: August 18, 2021; Accepted: April 28, 2022

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. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

The matrix Mittag-Leffler functions play a crucial role in several applications related to systems with fractional dynamics. These functions represent a generalization for fractional-order systems of the matrix exponential function involved in integer-order systems. Computational techniques for evaluating the matrix Mittag-Leffler functions are therefore of particular importance. In this study, a fuzzy system of differential equations with fractional derivatives was solved in terms of the matrix Mittag-Leffler functions. The matrix Mittag-Leffler function was evaluated based on the Jordan canonical form and the minimal polynomial of the matrix.

Keywords: Mittag-Leffler function, Fuzzy calculus, Fractional calculus, System of fuzzy fractional differential equations, Fractional differential equations

1. Introduction

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

Over the past few centuries, fractional differential (FDEs) have been successfully implemented in several mathematical models in the fields of biological science, chemistry, physics, and engineering [1]. The behavior of a number of physical systems can be properly described using a fractional-order system. Atanackovic and Stankovic [2] studied an FDE system that arises during lateral motion. The uniqueness, existence, and stability results for a system of FDEs have been presented in [3,4]. In the research community, several analytical and numerical techniques have been employed to determine solutions to a system of FDEs, such as the Laplace transform method [5,6], Adomian decomposition method [7], and exponential integrators [8].

A salient feature of the response of an FDE system is that the matrix Mittag-Leffler function serves as the basis of the solution, instead of the matrix exponential function as in the case of ordinary differential Owing to the difficulty in acquiring solutions in the form of fractional systems, numerical methods are often used to solve such fractional systems. Moret and Novati [9] applied the Krylov subspace method to evaluate the matrix Mittag-Leffler function. Matychyn and Onyshchenko [10] solved the Bagley-Tovik based on the Jordan canonical form.

Garrappa and N. Popolizio [11] investigated the computation of the Mittag-Leffler function and evaluated a three-parameter Mittag-Leffler function using the inverse Laplace transform method; the results have been presented in [12]. Duan and his colleague [13,14] employed different methods, such as the inverse Laplace transform method, Jordan canonical matrix method, and minimal polynomial method, for solving a system of FDEs, wherein the solutions were expressed in terms of the matrix Mittag-Leffler function.

Motivated by [13,14], in this study, we intend to apply the Jordan canonical method and the minimal polynomial method to a fuzzy system with fractional derivatives. In modeling real-world phenomena, the crisp quantities of a system of FDEs may cause imprecision and uncertainty. This uncertainty gave rise to several studies on the fuzzy systems of FDEs. We primarily aim to develop the Jordan canonical and minimal polynomial proposed by Duan [13] for computing the matrix Mittag-Leffler function in the solutions of a system of fuzzy FDEs.

Recently, several researchers have focused on fuzzy differential with fractional-order derivatives. Agarwal et al. [15] introduced a fuzzy FDE with a combination of the Hukuhara difference and Riemann-Liouville derivative. The existence of a solution of FDEs with uncertainty, involving the Riemann-Liouville operator, was proved by Arshad and Lupulescu [16] and Allahviranloo et al. [17]. However, the H-differentiable functions have an increasing length of support in the time variable. To overcome this issue, certain authors have considered the generalized Hukuhara differentiability (gH-differentiability), as in [18–22]. This derivative can enhance the set of fuzzy solutions and provide further insight into the behavior of fuzzy solutions. Balooch Shahriyar et al. [23] obtained solutions of a system of fuzzy fractional differential (SFFDEs) based on the method of eigenvalues and eigenvectors. The collocation method for discontinuous piecewise polynomial spaces has also been applied to the SFFDE in [24].

In our research, we compared our obtained solutions with the solutions obtained by the eigenvalue and eigenvector methods, presented in [23]. Our approach proves to be beneficial because it is non-differentiable, non-integral, and easy to implement using a computer owing to its matrix-based structure.

This remainder of this article is structured as follows: Section 2 provides the basic concepts of fuzzy and fractional calculus. Section 3 describes the solution procedure for fuzzy FDEs. Section 4 presents the computation of the matrix Mittag-Leffler function. In Section 5, several numerical examples are presented. Finally, Section 6 outlines the conclusions.

2. Basic Concepts

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

In this section, we review the basic definitions of fuzzy calculus, fractional calculus, and the matrix Mittag-Leffler function. Consider ℜ_F to be a collection of all fuzzy numbers on the real number ℜ.

Definition 2.1 [25]

The membership function for a fuzzy set w is defined by w : ℜ → [0, 1], and w(p) is denoted as the degree of membership of an element p in the fuzzy set w for each p ∈ ℜ. If w is a convex fuzzy set, then for each p, q ∈ ℜ and λ ∈ [0, 1], wehavew(λp+(1−λ)q) ≥ min{w(p), w(q)}. If w is a normal fuzzy set, then there exists a p ∈ ℜ such that w(p) = 1. The support for w is denoted by sup(w) and is defined by sup(w) = cl{p ∈ ℜ : w(p) > 0}. Moreover, w is upper semi-continuous.

Definition 2.2 [25,26]

If w is a fuzzy number, then it is defined by [w]^r = [w_r, w̄_r], which satisfies the following conditions:

(1) w_r is an increasing function, and w̄_r is a decreasing function such that w_r ≤ w̄_r.
(2) w_r and w̄_r are bounded and left-hand continuous functions in (0,1].
(3) w_r and w̄_r are bounded and right-hand continuous functions at r = 0.

Definition 2.3 [27]

Let w, y ∈ ℜ_F. If there exists a z ∈ ℜ_F such that w = y+z, then z is referred to as the H-difference between w and y and is denoted by w −_H y. The symbol −_H represents the H-difference, and w −_H y ≠ w + (−1)y.

Definition 2.4 [28]

Let I = [a, b]. For F : I → ℜ_F and p₀ ∈ I, if F is strongly H-differentiable at p₀, then there exists an element F′(p₀) ∈ ℜ_F such that

(1) H-differences F(p₀+ω)−_HF(p₀) and F(p₀)−_HF(p₀− ω) exist for eachω > 0 sufficiently close to 0; additionally, lim

limω→0F(p0+ω)-HF(p0)ω=limω→0F(p0)-HF(p0-ω)ω=F′(p0).
(2) H-differences F(p₀) −_H F(p₀ + ω) and F(p₀ − ω) −_H F(p₀) exist for eachω > 0 sufficiently close to 0; additionally,

limω→0F(p0)-HF(p0+ω)-ω=limω→0F(p0-ω)-HF(p0)-ω=F′(p0).

Definition 2.5 [29]

Let F : I → ℜ_F be a fuzzy function, and let F(p; r) = [F(p; r), F̄(p; r)] for r ∈ [0, 1].

(1) F is assumed to be (1)-type differentiable if F(p; r) and F̄(p; r) are differentiable functions, and F′(p;r)=[F′_(p;r),F′¯(p;r)]
(2) F is assumed to be (2)-type differentiable if F(p; r) and F̄(p; r) are differentiable functions, and F′(p;r)=[F′¯(p;r),F′_(p;r)]

Definition 2.6 [30,31]

Let F : I → ℜ_F and F ∈ C^F (I) ∩ L^F (I). The fuzzy Riemann-Liouville fractional integral of order α is described as follows:

Jα(F(p))=1Γ(α)∫apF(v)(p-v)1-αdv, 0<α≤1.

Definition 2.7 [31,32]

Let F : I → ℜ_F and F ∈ C^F (I) ∩ L^F (I). Then, F is Caputo’s H-differentiable at p if

(DCαF) (p)=1Γ(1-α)∫apF′(v)(p-v)αdv.0<α≤1.

In addition, F is Caputo [(1)-type - α] differentiable when F is (1)-type differentiable, and F is Caputo [(2)-type - α] differentiable when F is (2)-type differentiable, where C^F (I) and L^F (I) represent the space of all continuous and Lebesgue integrable fuzzy valued functions on I, respectively.

Definition 2.8

The Mittag–Leffler function is crucial for representing the solution of FDEs. The standard definition of the Mittag-Leffler function with one parameter and two parameters [33,34] is given as

Eα(z)=∑k=0∞zkΓ(αk+1), α>0, z∈C,Eα, β(z)=∑k=0∞zkΓ(αk+β), α>0, β>0, z∈C,

where Γ denotes a gamma function Γ(y)=∫0∞ty-1e-tdt.

Consider A ∈ C^nXn. Then, the matrix Mittag-Leffler function [9] is given as

(1)Eα, β(A)=∑k=0∞AkΓ(αk+β), α>0, β>0.

By setting β = 1 in Eq. (1), we obtain the one-parameter matrix Mittag-Leffler function as E_α(A). Moreover, by setting α = β = 1, the matrix Mittag-Leffler function becomes a matrix exponential, i.e.,

E1, 1(A)=∑k=0∞Akk!=eA.

3. Solution for a System of Fuzzy FDEs

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

We consider the following SFFDE:

(2)DtαX˜(t)=AX˜(t)+f˜(t), t>0,

where , A denotes the n-th order crisp matrix, f̃(t) = (f̃₁(t), f̃₂(t), …, f̃_n(t))^T denotes a continuous function, and the elements of f̃ (t) are fuzzy numbers. The fuzzy initial condition X̃ (0) = (x̃₁(0), x̃₂(0), …, x̃_n(0))^T is specified, and Dtα is the fuzzy Caputo’s fractional derivative, where 0 < α ≤ 1.

By operating on both sides of Eq. (2), we obtain

(3)X˜(t)=X˜(0)+AJαX˜(t)+Jαf˜(t),

where

X˜(0)=[X_(0;r),X¯(0;r)]=((x_1(0;r),x¯1(0;r)),(x_2(0;r),x¯2(0;r)), …,(x_n(0;r),x¯n(0;r)))T.

The parametric form of Eq. (3) is given as

X_(t;r)=X_(0;r)+AJαX_(t;r)+Jαf_(t;r),X¯(t;r)=X¯(0;r)+AJαX¯(t;r)+Jαf¯(t;r).

Let us denote the k^th approximate solution by ϕ_k (t; r) and ϕ̄_k(t; r), where the zeroth approximate solution is given by

ϕ_0(t;r)=X_(0;r);ϕ¯0(t;r)=X¯(0;r).

For k ≥ 1, we derive the following recursive formula:

(4)ϕ_k(t;r)=X_(0;r)+AJαϕ_k-1(t;r)+Jαf_(t;r),ϕ¯k(t;r)=X¯(0;r)+AJαϕ¯k-1(t;r)+Jαf¯(t;r).

Using Eq. (4), we can calculate the following:

ϕ_1(t;r)=X_(0;r)+AtαΓ(α+1)X_(0;r)+Jαf_(t;r),ϕ¯1(t;r)=X¯(0;r)+AtαΓ(α+1)X¯(0;r)+Jαf¯(t;r).

Similarly,

ϕ_2(t;r)=X_(0;r)+AtαΓ(α+1)X_(0;r)+A2t2αΓ(2α+1)X_(0;r)+Jαf_(t;r)+AJ2αf_(t;r),ϕ¯2(t;r)=X¯(0;r)+AtαΓ(α+1)X¯(0;r)+A2t2αΓ(2α+1)X¯(0;r)+Jαf¯(t;r)+AJ2αf¯(t;r),⋯ϕ_k(t;r)=∑i=0kAitiαΓ(iα+1)X_(0;r)+∑i=0kAiJ(i+1)αf_(t;r),ϕ¯k(t;r)=∑i=0kAitiαΓ(iα+1)X¯(0;r)+∑i=0kAiJ(i+1)αf¯(t;r).

Then, the formula for the fractional integral is

J(i+1)αf_(t;r)=tiα+α-1Γ(iα+α)*f_(t;r),

and

J(i+1)αf¯(t;r)=tiα+α-1Γ(iα+α)*f¯(t;r).

By applying the limit k → ∞ for ϕ^k (t; r) and ϕ̃_k(t; r), we obtain the solution in a series form:

(5)X_(t;r)=∑i=0∞AitiαΓ(iα+1)X⌣_(0;r)+∑i=0∞Aitiα+α-1Γ(iα+α)*f_(t;r),X¯(t;r)=∑i=0∞AitiαΓ(iα+1)X⌣¯(0;r)+∑i=0∞Aitiα+α-1Γ(iα+α)*f¯(t;r).

Using the matrix Mittag-Leffler function, we can express the solution as follows:

(6)X_(t;r)=Eα(Atα)X_(0;r)+tα-1Eα, α(Atα)*f_(t;r),X¯(t;r)=Eα(Atα)X¯(0;r)+tα-1Eα, α(Atα)*f¯(t;r).

where the convolution is defined as tα-1Eα, α(Atα)*f(t;r)=∫0tτα-1Eα, α(Aτα)f(t-τ;r)dτ. Then, the solution of Eq. (2) is of the form

(7)X˜(t)=Eα(Atα)X˜(0)+tα-1Eα, α(Atα)*f˜(t).

Theorem 3.1

If A has real distinct eigenvalues λ_i and the corresponding eigenvector ζ_i for i = 1, 2, …, n, the solution of the fuzzy fractional system (2) is a fuzzy number X˜(t)=∑i=1nc˜iEα(λitα)ζi, where f̃(t) = 0.

Proof

Consider X˜(t)=∑i=1nc˜iEα(λitα)ζi to be a solution to the fuzzy fractional system (2).

Suppose C̃ = [c̃₁, c̃₂, …, c̃_n], where c̃_i are fuzzy numbers, and [C˜]r=∏i=1n[c˜i]r. Then,

(8)X_(t;r) =min{∑i=1nciEα(λitα)ζi/ci∈[C˜]r} =∑i=1nciEα(λitα)ζi_,X¯(t;r) =∑i=1nciEα(λitα)ζi_ by Definition 7 in [23],

(9)X¯(t;r)=max{∑i=1nciEα(λitα)ζi/ci∈[C˜]r}=∑i=1nciEα(λitα)ζi¯=∑i=1nciEα(λitα)ζi¯.

Differentiating Eqs. (8) and (9), we obtain

DtαX_(t;r)=∑i=1nciλiEα(λitα)ζi_,

and

DtαX¯(t;r)=∑i=1nciλiEα(λitα)ζi¯.

Because λ_i and ζ_i are the eigenvalues and corresponding eigenvectors of matrix A, respectively, Aζ_i=λζ_i for i = 1, 2, …, n. Then, we have

DtαX_(t;r)=A∑i=1nciEα(λitα)ζi_=AX_(t;r),DtαX¯(t;r)=A∑i=1nciEα(λitα)ζi¯=AX¯(t;r).

i.e.,

(DtαX_(t;r),DtαX¯(t;r))=A(X_(t;r),X¯(t;r)),DtαX˜(t)=AX˜(t).

This completes the proof.

Theorem 3.2

If A has repeated eigenvalues λ_i with multiplicity, and k and ζ_i, i = 1, 2, …, n are the corresponding eigenvectors, then the solution of the fuzzy fractional system (2) is a fuzzy number.

(10)X˜(t)=∑i=1kc˜i(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)+∑i=k+1nc˜iEα(λitα)ζi,

where f̃(t) = 0.

Proof

Consider Eq. (10) as the solution of the fuzzy fractional system (2). Suppose C̃ = [c̃₁, c̃₂, …, c̃_n], where c̃_i are fuzzy numbers, and [C˜]r=∏i=1n[c˜i]r. Then,

(11)X_(t;r)=min{∑i=1kci(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)+∑i=k+1nciEα(λitα)ζi/ci∈[C˜]r}=∑i=1kci(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)+∑i=k+1nciEα(λitα)ζi_=∑i=1kci(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)_+∑i=k+1nciEα(λitα)ζi_,

(12)X¯(t;r)=max{∑i=1kci(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)+∑i=k+1nciEα(λitα)ζi/ci∈[C˜]r}=∑i=1kci(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)+∑i=k+1nciEα(λitα)ζi¯=∑i=1kci(Eα(k)(λitα) (tα(A-λiI)ζi)+Eα(λitα)ζi)¯+∑i=k+1nciEα(λitα)ζi¯.

Differentiating Eqs. (11) and (12), we obtain

DtαX_(t;r)=∑i=1kci(λiEα(k)(λitα) (tα(A-λiI)ζi)+Eα(k)(λitα) (A-λiI)ζi+λiEα(λitα)ζi)_+∑i=k+1nciλiEα(λitα)ζi_,DtαX¯(t;r)=∑i=1kci(λiEα(k)(λitα) (tα(A-λiI)ζi)+Eα(k)(λitα) (A-λiI)ζi+λiEα(λitα)ζi)¯+∑i=k+1nciλiEα(λitα)ζi¯

Because λ_i and ζ_i are eigenvalues and the corresponding eigenvectors of matrix A, Aζ_i = λζ_i for i = 1, 2, …, n. Then, we have

DtaX_(t;r)=AX_(t;r),DtaX¯(t;r)=AX¯(t;r),

i.e.,

(DtaX_(t;r),DtaX¯(t;r))=A(X_(t;r),X¯(t;r)),DtaX˜(t)=AX˜(t).

This completes the proof.

4. Calculation of the Matrix Mittag-Leffler Function

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

The solution of the SFFDEs consists of the matrix Mittag-Leffler function E_α(At^α). In this section, we obtain this matrix Mittag-Leffler function using the Jordan canonical method and the minimal polynomial method.

4.1 Matrix of the Jordan Canonical Method

Consider that the matrix of the Jordan canonical for A with an order of n×n is J. J = diag(J₁, J₂, …, J_k) and A = PJP⁻¹, where

(13)Ji=[λi1 λi1 ⋱⋱ ⋱1 λi]ni×ni, i=1, 2, …, k,

and ∑i=1kni=n. Thus, we have

(14)Eα(Atα)=PEα(Jtα)P-1=Pdiag(Eα(J1tα), …,Eα(Jktα))P-1,

where E_α(J_it^α) are upper triangular matrices.

Eα(Jitα)=[Eα(λitα)tαEα′(λitα)…t(ni-1)αEα(ni-1)(λitα)(ni-1)! Eα(λitα)…t(ni-2)αEα(ni-2)(λitα)(ni-2)! ⋱⋮ ⋱tαEα′(λitα) Eα(λitα)],

where Eαk(λitα)=dkEα(z)dzk∣z=λitα

4.2 Minimal Polynomial Method

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

Consider that the minimal polynomial of the nth order matrix A is m(λ) = (λ − λ₁)ⁿ^₁ (λ − λ₂)ⁿ^₂ … (λ − λ_k)n_^k, where ∑i=1kni=n.

According to the theory of matrix analysis [35], we can express E_α(At^α) as a matrix polynomial of degree n − 1, i.e.,

(15)Eα(Atα)=a0(t)I+a1(t)A+⋯+am-1(t)Am-1.

Eq. (15) holds if and only if E_α(λt^α) and the polynomial ψ(λ, t) = a₀(t)+a₁(t)λ+· · ·+a_m₋₁(t)λ^m⁻¹ are consistent in the spectrum of matrix A; that is,

(16)djψ(λ, t)dλj∣λ=λi=tjαEα(j)(λitα),i=1, 2, …k, j=0, 1, …ni-1.

We can obtain a_i(t) by solving Eq. (16).

5. Numerical Examples

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

5.1 Example

We consider the following SFFDE:

(17)DtaX˜(t)=AX˜(t), A=(2-14-3).

With an initial condition,

X˜(0)=(x1_(0)x2_(0)x1¯(0)x2¯(0))=(r+12+r3-r5-r),

where λ₁ = 1 and λ₂ = −2 denote the eigenvalues of the matrix A.

By applying the Jordan canonical method, we have A = PJP⁻¹, where

(18)P=(1114) and J=(100-2),Eα(Atα)=PEα(Atα)P-1=P(Eα(tα)00Eα(-2tα))P-1=13 (4-14-1) Eα(tα)+13 (-11-44) Eα(-2tα).

We apply the minimal polynomial method as follows:

Next, we consider the minimal polynomial of A as m(λ) = (λ − 1)(λ + 2). Further, we consider E_α(At^α) = a₀(t)I₂ + a₁(t)A, where I₂ represents a second-order identity matrix. Thus,

a0(t)+a1(t)=Eα(tα),a0(t)-2a1(t)=Eα(-2tα).

By solving the abovementioned system, we obtain

a0(t)=23Eα(tα)+13Eα(-2tα),a1(t)=13Eα(tα)-13Eα(-2tα).

In addition, the matrix Mittag-Leffler function has the form

(19)Eα(Atα)=[23Eα(tα)+13Eα(-2tα)]I2+[13Eα(tα)-13Eα(-2tα)]A=13 (4-14-1) Eα(tα)+13 (-11-44) Eα(-2tα).

The results presented in Eqs. (18) and (19) are the same.

The solution of SFFDE (17) parameterized by the order α is

X˜(t)=Eα(Atα)X˜(0).

Then,

X_(t;r)=(r+23) (11) Eα(tα)+13 (14) Eα(-2tα),X¯(t;r)=(73-r) (11) Eα(tα)+23 (14) Eα(-2tα).

The above results are consistent with those obtained by the eigenvalue and eigenvector methods in [23, Eg. 1].

5.2 Example

Consider the following SFFDE:

(20)DtαX˜(t)=AX˜(t), A=(11121-10-11).

With an initial condition,

X˜(0)=(x1_(0)x2_(0)x3_(0)x1¯(0)x2¯(0)x3¯(0))=(0.75+0.25r1.5+0.5r3.75+0.25r1.25-0.25r2.5-0.5r4.25-0.25r).

Subsequently, λ₁ = −1 and λ₂ = λ₃ = 2 denote the eigenvalues of matrix A. By applying the Jordan canonical method, we obtain A = PJP⁻¹, where

(21)P=(-3014102-11), J=(-100021002),Eα(Atα)=PEα(Atα)P-1=P(Eα(-tα)000Eα(2tα)tαEα′(2tα)00Eα(2tα))P-1=19 (3-3-3-444-222) Eα(-tα)+19 (63345-42-27) Eα(2tα)+19 (000633-6-3-3)tαEα′(2tα).

Further, we apply the minimal polynomial method as follows:

The minimal polynomial of A is m(λ) = (λ + 1)(λ − 2)² and E_α(At^α) = a₀(t)I₃ + a₁(t)A + a₂(t)A². Thus,

a0(t)-a1(t)+a2(t)=Eα(-tα),a0(t)+2a1(t)+4a2(t)=Eα(2tα),a1(t)+4a2(t)=tαEα′(2tα).

We now solve the system to obtain

a0(t)=49Eα(-tα)+59Eα(2tα)-69tαEα′(2tα),a1(t)=-49Eα(-tα)+49Eα(2tα)-39tαEα′(2tα),a2(t)=19Eα(-tα)-19Eα(2tα)+39tαEα′(2tα).

Then, the matrix Mittag-Leffler function assumes the form

(22)Eα(Atα)= [49Eα(-tα)+59Eα(2tα)-69tαEα′(2tα)]I3 +[-49Eα(-tα)+49Eα(2tα)-39tαEα′(2tα)]A +[19Eα(-tα)-19Eα(2tα)+39tαEα′(2tα)]A2= 19 (3-3-3-444222) Eα(-tα)+19 (63345-42-27) Eα(2tα) +19 (000633-6-3-3)tαEα′(2tα).

The results presented in Eqs. (21) and (22) are the same. The solution of SFFDE (20) parameterized by the order α is

X˜(t)=Eα(Atα)X˜(0).

Then,

X_(t;r)= 19 (-13.5-1.5r18+2r9+r) Eα(-tα) +19 (20.25+3.75r-4.5+2.5r24.75+1.25r) Eα(2tα) +19 (020.25+3.75r-20.25-3.75r)tαEα′(2tα),X¯(t;r)= 19 (-16.5+1.5r22-2r11-r) Eα(-tα) +19 (27.75-3.75r0.5-2.5r27.25-1.25r) Eα(2tα) +19 (027.75-3.75r-27.75+3.75r)tαEα′(2tα).

The solutions presented above can be rewritten in the following form:

X_(t;r)= 19(4.5+0.5r) (-342) Eα(-tα) +19(-4.5+2.5r) (01-1) Eα(2tα) +19(20.25+3.75r) [(01-1)tαEα′(2tα)+(101) Eα(2tα)],X¯(t;r)= 19(5.5-0.5r) (-342) Eα(-tα) +19(0.5-2.5r) (01-1) Eα(2tα) +19(27.75-3.75r) [(01-1)tαEα′(2tα)+(101) Eα(2tα)].

The above results are consistent with those obtained by the eigenvalue and eigenvector methods in [23, Eg. 2].

Figure 1(a) and 1(b) present the solution curves of Example 1 for different values of α. Figure 2(a)–2(c) present the solution curves of Example 2 for different values of α, wherein the purple, blue, and red lines represent α = 0.6, α = 0.8, and α = 1.0, respectively. In the examples above, the results indicate that the solutions obtained by the proposed method are similar to those obtained by the eigenvalue and eigenvector method in [23].

6. Conclusion

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

This study primarily aimed to develop a solution for a system of fuzzy FDEs in terms of the matrix Mittag-Leffler functions via two methods: the Jordan canonical method and the minimal polynomial method. The effectiveness, ability, and simplicity of the proposed techniques were demonstrated using numerical examples. The numerical and graphical results revealed that the solutions obtained by the proposed methods were consistent with the solutions obtained by the eigenvalue and eigenvector method. The solutions were plotted using MATLAB.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

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

Figures

Go to

Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

Fig. 1.

Solutions x̃₁(t) and x̃₂(t) of Example 1 for different values of (a) α = 0.6, (b) α = 0.8, (c) α = 1.0 at t = 1.

Fig. 2.

Solutions x̃₁(t), x̃₂(t), and x̃₃(t) of Example 2 for different values of (a) α = 0.6, (b) α = 0.8, (c) α = 1.0 at t = 1.

References

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

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Atanackovic, TM, and Stankovic, B (2004). On a system of differential equations with fractional derivatives arising in rod theory. Journal of Physics A: Mathematical and General. 37, 1241-1250. https://doi.org/10.1088/0305-4470/37/4/012
Daftardar-Gejji, V, and Babakhani, A (2004). Analysis of a system of fractional differential equations. Journal of Mathematical Analysis and Applications. 293, 511-522. https://doi.org/10.1016/j.jmaa.2004.01.013
Deng, W, Li, C, and Lu, J (2007). Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics. 48, 409-416. https://doi.org/10.1007/s11071-006-9094-0
Charef, A, and Boucherma, D (2011). Analytical solution of the linear fractional system of commensurate order. Computers & Mathematics with Applications. 62, 4415-4428. https://doi.org/10.1016/j.camwa.2011.10.017
Duan, JS, Fu, SZ, and Wang, Z (2013). Solution of linear system of fractional differential equations. Pacific Journal of Applied Mathematics. 5, 93-106.
Daftardar-Gejji, V, and Jafari, H (2005). Adomian decomposition: a tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications. 301, 508-518. https://doi.org/10.1016/j.jmaa.2004.07.039
Garrappa, R (2013). Exponential integrators for time–fractional partial differential equations. The European Physical Journal Special Topics. 222, 1915-1927. https://doi.org/10.1140/epjst/e2013-01973-1
Moret, I, and Novati, P (2011). On the convergence of Krylov subspace methods for matrix Mittag–Leffler functions. SIAM Journal on Numerical Analysis. 49, 2144-2164. https://doi.org/10.1137/080738374
Matychyn, I, and Onyshchenko, V (2018). Matrix Mittag Leffler function in fractional systems and its computation. Bulletin of the Polish Academy of Sciences: Technical Sciences. 66, 495-500. https://doi.org/10.24425/124266
Garrappa, R, and Popolizio, N (2018). Computing the matrix Mittag-Leffler function with applications to fractional calculus. Journal of Scientific Computing. 77, 129-153. https://doi.org/10.1007/s10915-018-0699-5
Garrappa, R (2015). Numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM Journal on Numerical Analysis. 53, 1350-1369. https://doi.org/10.1137/140971191
Duan, J (2018). System of linear fractional differential equations and the Mittag-Leffler functions with matrix variable. Journal of Physics: Conference Series. 1053. article no. 012032
Duan, J, and Chen, L (2018). Solution of fractional differential equation systems and computation of matrix Mittag–Leffler functions. Symmetry. 10. article no. 503
Agarwal, RP, Lakshmikantham, V, and Nieto, JJ (2010). On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods & Applications. 72, 2859-2862. https://doi.org/10.1016/j.na.2009.11.029
Arshad, S, and Lupulescu, V (2011). On the fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods & Applications. 74, 3685-3693. https://doi.org/10.1016/j.na.2011.02.048
Allahviranloo, T, Salahshour, S, and Abbasbandy, S (2012). Explicit solutions of fractional differential equations with uncertainty. Soft Computing. 16, 297-302. https://doi.org/10.1007/s00500-011-0743-y
Allahviranloo, T, Armand, A, and Gouyandeh, Z (2014). Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. Journal of Intelligent & Fuzzy Systems. 26, 1481-1490. https://doi.org/10.3233/IFS-130831
Van Hoa, N (2015). Fuzzy fractional functional differential equations under Caputo gH-differentiability. Communications in Nonlinear Science and Numerical Simulation. 22, 1134-1157. https://doi.org/10.1016/j.cnsns.2014.08.006
Van Hoa, N (2015). Fuzzy fractional functional integral and differential equations. Fuzzy Sets and Systems. 280, 58-90. https://doi.org/10.1016/j.fss.2015.01.009
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Long, HV, Son, NTK, and Tam, HTT (2017). The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability. Fuzzy Sets and Systems. 309, 35-63. https://doi.org/10.1016/j.fss.2016.06.018
Balooch Shahriyar, MR, Ismail, F, Aghabeigi, S, Ahmadian, A, and Salahshour, S (2013). An eigenvalue-eigenvector method for solving a system of fractional differential equations with uncertainty. Mathematical Problems in Engineering. 2013. article no. 579761
Alijani, Z, Baleanu, D, Shiri, B, and Wu, GC (2020). Spline collocation methods for systems of fuzzy fractional differential equations. Chaos, Solitons & Fractals. 131. article no. 109510
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Chalco-Cano, Y, and Roman-Flores, H (2008). On new solutions of fuzzy differential equations. Chaos, Solitons & Fractals. 38, 112-119. https://doi.org/10.1016/j.chaos.2006.10.043
Abasbandy, S, Allahviranloo, T, Shahryari, MB, and Salahshour, S (2012). Fuzzy local fractional differential equation. International Journal of Industrial Mathematics. 4, 231-245.
Allahviranloo, T. (2021) . Fuzzy Fractional Differential Operators and Equations. https://doi.org/10.1007/978-3-030-51272-9
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Biographies

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Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion
Conflict of Interest
Figures
References
Biographies

V. Padmapriya is a Ph.D. research scholar studying at the Division of Mathematics, Vellore Institute of Technology-Chennai Campus, India. She is currently working as an assistant professor in the Department of Statistics, New prince Shri Bhavani Arts and Science college, Chennai, India. Her research interests include fuzzy differential and fuzzy fractional differential

E-mail: v.padmapriya2015@vit.ac.in

M. Kaliyappan is working as a professor of Mathematics at the Division of Mathematics, School of Advanced Sciences, VIT Chennai, India. He has over 23 years of experience in teaching and research. His research interests include approximation theory, numerical computing, differential equations, fuzzy differential equations, fractional differential equations, fuzzy fractional differential equations, and optimization theory.

E-mail: kaliyappan.m@vit.ac.in

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 144-154

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.144

Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

V. Padmapriya^1,² and M. Kaliyappan³

Correspondence to:V. Padmapriya (v.padmapriya2015@vit.ac.in)

Received: August 18, 2021; Revised: August 18, 2021; Accepted: April 28, 2022

Abstract

Keywords: Mittag-Leffler function, Fuzzy calculus, Fractional calculus, System of fuzzy fractional differential equations, Fractional differential equations

1. Introduction

2. Basic Concepts

Definition 2.1 [25]

Definition 2.2 [25,26]

If w is a fuzzy number, then it is defined by [w]^r = [w_r, w̄_r], which satisfies the following conditions:

(1) w_r is an increasing function, and w̄_r is a decreasing function such that w_r ≤ w̄_r.
(2) w_r and w̄_r are bounded and left-hand continuous functions in (0,1].
(3) w_r and w̄_r are bounded and right-hand continuous functions at r = 0.

Definition 2.3 [27]

Definition 2.4 [28]

Let I = [a, b]. For F : I → ℜ_F and p₀ ∈ I, if F is strongly H-differentiable at p₀, then there exists an element F′(p₀) ∈ ℜ_F such that

(1) H-differences F(p₀+ω)−_HF(p₀) and F(p₀)−_HF(p₀− ω) exist for eachω > 0 sufficiently close to 0; additionally, lim

$\begin{array}{l} lim_{ω \to 0} \frac{F (p_{0} + ω) -_{H} F (p_{0})}{ω} \\ = lim_{ω \to 0} \frac{F (p_{0}) -_{H} F (p_{0} - ω)}{ω} \\ = F^{'} (p_{0}) . \end{array}$
(2) H-differences F(p₀) −_H F(p₀ + ω) and F(p₀ − ω) −_H F(p₀) exist for eachω > 0 sufficiently close to 0; additionally,

\begin{array}{l} lim_{ω \to 0} \frac{F (p_{0}) -_{H} F (p_{0} + ω)}{- ω} \\ = lim_{ω \to 0} \frac{F (p_{0} - ω) -_{H} F (p_{0})}{- ω} \\ = F^{'} (p_{0}) . \end{array}

Definition 2.5 [29]

Let F : I → ℜ_F be a fuzzy function, and let F(p; r) = [F(p; r), F̄(p; r)] for r ∈ [0, 1].

(1) F is assumed to be (1)-type differentiable if F(p; r) and F̄(p; r) are differentiable functions, and $F^{'} (p; r) = [\underline{F^{'}} (p; r), \bar{F^{'}} (p; r)]$
(2) F is assumed to be (2)-type differentiable if F(p; r) and F̄(p; r) are differentiable functions, and $F^{'} (p; r) = [\bar{F^{'}} (p; r), \underline{F^{'}} (p; r)]$

Definition 2.6 [30,31]

Let F : I → ℜ_F and F ∈ C^F (I) ∩ L^F (I). The fuzzy Riemann-Liouville fractional integral of order α is described as follows:

J^{α} (F (p)) = \frac{1}{Γ (α)} \int_{a}^{p} \frac{F (v)}{{(p - v)}^{1 - α}} d v, 0 < α \leq 1.

Definition 2.7 [31,32]

Let F : I → ℜ_F and F ∈ C^F (I) ∩ L^F (I). Then, F is Caputo’s H-differentiable at p if

({{}^{C}D}^{α} F) (p) = \frac{1}{Γ (1 - α)} \int_{a}^{p} \frac{F^{'} (v)}{{(p - v)}^{α}} d v .0 < α \leq 1.

Definition 2.8

The Mittag–Leffler function is crucial for representing the solution of FDEs. The standard definition of the Mittag-Leffler function with one parameter and two parameters [33,34] is given as

\begin{array}{l} E_{α} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + 1)}, α > 0, z \in C, \\ E_{α, β} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + β)}, α > 0, β > 0, z \in C, \end{array}

where Γ denotes a gamma function $Γ (y) = \int_{0}^{\infty} t^{y - 1} e^{- t} d t$ .

Consider A ∈ C^nXn. Then, the matrix Mittag-Leffler function [9] is given as

(1)

E_{α, β} (A) = \sum_{k = 0}^{\infty} \frac{A^{k}}{Γ (α k + β)}, α > 0, β > 0.

E_{1, 1} (A) = \sum_{k = 0}^{\infty} \frac{A^{k}}{k!} = e^{A} .

3. Solution for a System of Fuzzy FDEs

We consider the following SFFDE:

(2)

D_{t}^{α} \tilde{X} (t) = A \tilde{X} (t) + \tilde{f} (t), t > 0,

where , A denotes the n-th order crisp matrix, f̃(t) = (f̃₁(t), f̃₂(t), …, f̃_n(t))^T denotes a continuous function, and the elements of f̃ (t) are fuzzy numbers. The fuzzy initial condition X̃ (0) = (x̃₁(0), x̃₂(0), …, x̃_n(0))^T is specified, and $D_{t}^{α}$ is the fuzzy Caputo’s fractional derivative, where 0 < α ≤ 1.

By operating on both sides of Eq. (2), we obtain

(3)

\tilde{X} (t) = \tilde{X} (0) + A J^{α} \tilde{X} (t) + J^{α} \tilde{f} (t),

where

\begin{array}{l} \tilde{X} (0) = [\underline{X} (0; r), \bar{X} (0; r)] \\ = {(({\underline{x}}_{1} (0; r), {\bar{x}}_{1} (0; r)), ({\underline{x}}_{2} (0; r), {\bar{x}}_{2} (0; r)), \dots, ({\underline{x}}_{n} (0; r), {\bar{x}}_{n} (0; r)))}^{T} . \end{array}

The parametric form of Eq. (3) is given as

\begin{array}{l} \underline{X} (t; r) = \underline{X} (0; r) + A J^{α} \underline{X} (t; r) + J^{α} \underline{f} (t; r), \\ \bar{X} (t; r) = \bar{X} (0; r) + A J^{α} \bar{X} (t; r) + J^{α} \bar{f} (t; r) . \end{array}

Let us denote the k^th approximate solution by ϕ_k (t; r) and ϕ̄_k(t; r), where the zeroth approximate solution is given by

{\underline{ϕ}}_{0} (t; r) = \underline{X} (0; r); {\bar{ϕ}}_{0} (t; r) = \bar{X} (0; r) .

For k ≥ 1, we derive the following recursive formula:

(4)

\begin{array}{l} {\underline{ϕ}}_{k} (t; r) = \underline{X} (0; r) + A J^{α} {\underline{ϕ}}_{k - 1} (t; r) + J^{α} \underline{f} (t; r), \\ {\bar{ϕ}}_{k} (t; r) = \bar{X} (0; r) + A J^{α} {\bar{ϕ}}_{k - 1} (t; r) + J^{α} \bar{f} (t; r) . \end{array}

Using Eq. (4), we can calculate the following:

\begin{array}{l} {\underline{ϕ}}_{1} (t; r) = \underline{X} (0; r) + \frac{A t^{α}}{Γ (α + 1)} \underline{X} (0; r) + J^{α} \underline{f} (t; r), \\ {\bar{ϕ}}_{1} (t; r) = \bar{X} (0; r) + \frac{A t^{α}}{Γ (α + 1)} \bar{X} (0; r) + J^{α} \bar{f} (t; r) . \end{array}

Similarly,

\begin{array}{l} {\underline{ϕ}}_{2} (t; r) = \underline{X} (0; r) + \frac{A t^{α}}{Γ (α + 1)} \underline{X} (0; r) + \frac{A^{2} t^{2 α}}{Γ (2 α + 1)} \underline{X} (0; r) + J^{α} \underline{f} (t; r) + A J^{2 α} \underline{f} (t; r), \\ {\bar{ϕ}}_{2} (t; r) = \bar{X} (0; r) + \frac{A t^{α}}{Γ (α + 1)} \bar{X} (0; r) + \frac{A^{2} t^{2 α}}{Γ (2 α + 1)} \bar{X} (0; r) + J^{α} \bar{f} (t; r) + A J^{2 α} \bar{f} (t; r), \\ \dots \\ {\underline{ϕ}}_{k} (t; r) = \sum_{i = 0}^{k} \frac{A^{i} t^{i α}}{Γ (i α + 1)} \underline{X} (0; r) + \sum_{i = 0}^{k} A^{i} J^{(i + 1) α} \underline{f} (t; r), \\ {\bar{ϕ}}_{k} (t; r) = \sum_{i = 0}^{k} \frac{A^{i} t^{i α}}{Γ (i α + 1)} \bar{X} (0; r) + \sum_{i = 0}^{k} A^{i} J^{(i + 1) α} \bar{f} (t; r) . \end{array}

Then, the formula for the fractional integral is

J^{(i + 1) α} \underline{f} (t; r) = \frac{t^{i α + α - 1}}{Γ (i α + α)} * \underline{f} (t; r),

and

J^{(i + 1) α} \bar{f} (t; r) = \frac{t^{i α + α - 1}}{Γ (i α + α)} * \bar{f} (t; r) .

By applying the limit k → ∞ for ϕ^k (t; r) and ϕ̃_k(t; r), we obtain the solution in a series form:

(5)

\begin{array}{l} \underline{X} (t; r) = \sum_{i = 0}^{\infty} \frac{A^{i} t^{i α}}{Γ (i α + 1)} \underline{\underset{⌣}{X}} (0; r) + \sum_{i = 0}^{\infty} \frac{A^{i} t^{i α + α - 1}}{Γ (i α + α)} * \underline{f} (t; r), \\ \bar{X} (t; r) = \sum_{i = 0}^{\infty} \frac{A^{i} t^{i α}}{Γ (i α + 1)} \bar{\underset{⌣}{X}} (0; r) + \sum_{i = 0}^{\infty} \frac{A^{i} t^{i α + α - 1}}{Γ (i α + α)} * \bar{f} (t; r) . \end{array}

Using the matrix Mittag-Leffler function, we can express the solution as follows:

(6)

\begin{array}{l} \underline{X} (t; r) = E_{α} (A t^{α}) \underline{X} (0; r) + t^{α - 1} E_{α, α} (A t^{α}) * \underline{f} (t; r), \\ \bar{X} (t; r) = E_{α} (A t^{α}) \bar{X} (0; r) + t^{α - 1} E_{α, α} (A t^{α}) * \bar{f} (t; r) . \end{array}

where the convolution is defined as $t^{α - 1} E_{α, α} (A t^{α}) * f (t; r) = \int_{0}^{t} τ^{α - 1} E_{α, α} (A τ^{α}) f (t - τ; r) d τ$ . Then, the solution of Eq. (2) is of the form

(7)

\tilde{X} (t) = E_{α} (A t^{α}) \tilde{X} (0) + t^{α - 1} E_{α, α} (A t^{α}) * \tilde{f} (t) .

Theorem 3.1

If A has real distinct eigenvalues λ_i and the corresponding eigenvector ζ_i for i = 1, 2, …, n, the solution of the fuzzy fractional system (2) is a fuzzy number $\tilde{X} (t) = \sum_{i = 1}^{n} {\tilde{c}}_{i} E_{α} (λ_{i} t^{α}) ζ_{i}$ , where f̃(t) = 0.

Proof

Consider $\tilde{X} (t) = \sum_{i = 1}^{n} {\tilde{c}}_{i} E_{α} (λ_{i} t^{α}) ζ_{i}$ to be a solution to the fuzzy fractional system (2).

Suppose C̃ = [c̃₁, c̃₂, …, c̃_n], where c̃_i are fuzzy numbers, and ${[\tilde{C}]}_{r} = \prod_{i = 1}^{n} {[{\tilde{c}}_{i}]}_{r}$ . Then,

(8)

\begin{array}{l} \underline{X} (t; r) & = m i n {\sum_{i = 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i} / c_{i} \in {[\tilde{C}]}_{r}} \\ = \underline{\sum_{i = 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}}, \\ \bar{X} (t; r) & = \sum_{i = 1}^{n} \underline{c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} by Definition 7 in [23], \end{array}

(9)

\begin{array}{l} \bar{X} (t; r) = m a x {\sum_{i = 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i} / c_{i} \in {[\tilde{C}]}_{r}} \\ = \bar{\sum_{i = 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} \\ = \sum_{i = 1}^{n} \bar{c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} . \end{array}

Differentiating Eqs. (8) and (9), we obtain

D_{t}^{α} \underline{X} (t; r) = \sum_{i = 1}^{n} \underline{c_{i} λ_{i} E_{α} (λ_{i} t^{α}) ζ_{i}},

and

D_{t}^{α} \bar{X} (t; r) = \sum_{i = 1}^{n} \bar{c_{i} λ_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} .

Because λ_i and ζ_i are the eigenvalues and corresponding eigenvectors of matrix A, respectively, Aζ_i=λζ_i for i = 1, 2, …, n. Then, we have

\begin{array}{l} D_{t}^{α} \underline{X} (t; r) = A \sum_{i = 1}^{n} \underline{c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} = A \underline{X} (t; r), \\ D_{t}^{α} \bar{X} (t; r) = A \sum_{i = 1}^{n} \bar{c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} = A \bar{X} (t; r) . \end{array}

i.e.,

\begin{array}{l} (D_{t}^{α} \underline{X} (t; r), D_{t}^{α} \bar{X} (t; r)) = A (\underline{X} (t; r), \bar{X} (t; r)), \\ D_{t}^{α} \tilde{X} (t) = A \tilde{X} (t) . \end{array}

This completes the proof.

Theorem 3.2

If A has repeated eigenvalues λ_i with multiplicity, and k and ζ_i, i = 1, 2, …, n are the corresponding eigenvectors, then the solution of the fuzzy fractional system (2) is a fuzzy number.

(10)

\tilde{X} (t) = \sum_{i = 1}^{k} {\tilde{c}}_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i}) + \sum_{i = k + 1}^{n} {\tilde{c}}_{i} E_{α} (λ_{i} t^{α}) ζ_{i},

where f̃(t) = 0.

Proof

Consider Eq. (10) as the solution of the fuzzy fractional system (2). Suppose C̃ = [c̃₁, c̃₂, …, c̃_n], where c̃_i are fuzzy numbers, and ${[\tilde{C}]}_{r} = \prod_{i = 1}^{n} {[{\tilde{c}}_{i}]}_{r}$ . Then,

(11)

\begin{array}{l} \underline{X} (t; r) = m i n {\sum_{i = 1}^{k} c_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i}) + \sum_{i = k + 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i} / c_{i} \in {[\tilde{C}]}_{r}} \\ = \underline{\sum_{i = 1}^{k} c_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i}) + \sum_{i = k + 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} \\ = \sum_{i = 1}^{k} \underline{c_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i})} + \sum_{i = k + 1}^{n} \underline{c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}}, \end{array}

(12)

\begin{array}{l} \bar{X} (t; r) = m a x {\sum_{i = 1}^{k} c_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i}) + \sum_{i = k + 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i} / c_{i} \in {[\tilde{C}]}_{r}} \\ = \bar{\sum_{i = 1}^{k} c_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i}) + \sum_{i = k + 1}^{n} c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} \\ = \sum_{i = 1}^{k} \bar{c_{i} (E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α} (λ_{i} t^{α}) ζ_{i})} + \sum_{i = k + 1}^{n} \bar{c_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} . \end{array}

Differentiating Eqs. (11) and (12), we obtain

\begin{array}{l} D_{t}^{α} \underline{X} (t; r) = \sum_{i = 1}^{k} \underline{c_{i} (λ_{i} E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α}^{(k)} (λ_{i} t^{α}) (A - λ_{i} I) ζ_{i} + λ_{i} E_{α} (λ_{i} t^{α}) ζ_{i})} + \sum_{i = k + 1}^{n} \underline{c_{i} λ_{i} E_{α} (λ_{i} t^{α}) ζ_{i}}, \\ D_{t}^{α} \bar{X} (t; r) = \sum_{i = 1}^{k} \bar{c_{i} (λ_{i} E_{α}^{(k)} (λ_{i} t^{α}) (t^{α} (A - λ_{i} I) ζ_{i}) + E_{α}^{(k)} (λ_{i} t^{α}) (A - λ_{i} I) ζ_{i} + λ_{i} E_{α} (λ_{i} t^{α}) ζ_{i})} + \sum_{i = k + 1}^{n} \bar{c_{i} λ_{i} E_{α} (λ_{i} t^{α}) ζ_{i}} \end{array}

Because λ_i and ζ_i are eigenvalues and the corresponding eigenvectors of matrix A, Aζ_i = λζ_i for i = 1, 2, …, n. Then, we have

\begin{array}{l} D_{t}^{a} \underline{X} (t; r) = A \underline{X} (t; r), \\ D_{t}^{a} \bar{X} (t; r) = A \bar{X} (t; r), \end{array}

i.e.,

\begin{array}{l} (D_{t}^{a} \underline{X} (t; r), D_{t}^{a} \bar{X} (t; r)) = A (\underline{X} (t; r), \bar{X} (t; r)), \\ D_{t}^{a} \tilde{X} (t) = A \tilde{X} (t) . \end{array}

This completes the proof.

4. Calculation of the Matrix Mittag-Leffler Function

4.1 Matrix of the Jordan Canonical Method

Consider that the matrix of the Jordan canonical for A with an order of n×n is J. J = diag(J₁, J₂, …, J_k) and A = PJP⁻¹, where

(13)

J_{i} = {[\begin{matrix} λ_{i} & 1 \\ λ_{i} & 1 \\ ⋱ & ⋱ \\ ⋱ & 1 \\ λ_{i} \end{matrix}]}_{n_{i} \times n_{i}}, i = 1, 2, \dots, k,

and $\sum_{i = 1}^{k} n_{i} = n$ . Thus, we have

(14)

\begin{array}{l} E_{α} (A t^{α}) = P E_{α} (J t^{α}) P^{- 1} \\ = P d i a g (E_{α} (J_{1} t^{α}), \dots, E_{α} (J_{k} t^{α})) P^{- 1}, \end{array}

where E_α(J_it^α) are upper triangular matrices.

E_{α} (J_{i} t^{α}) = [\begin{matrix} E_{α} (λ_{i} t^{α}) & t^{α} E_{α}^{'} (λ_{i} t^{α}) & \dots & \frac{t^{(n_{i} - 1) α} E_{α}^{(n_{i} - 1)} (λ_{i} t^{α})}{(n_{i} - 1)!} \\ E_{α} (λ_{i} t^{α}) & \dots & \frac{t^{(n_{i} - 2) α} E_{α}^{(n_{i} - 2)} (λ_{i} t^{α})}{(n_{i} - 2)!} \\ ⋱ & ⋮ \\ ⋱ & t^{α} E_{α}^{'} (λ_{i} t^{α}) \\ E_{α} (λ_{i} t^{α}) \end{matrix}],

where $E_{α}^{k} (λ_{i} t^{α}) = {\frac{d^{k} E_{α} (z)}{d z^{k}} ∣}_{z = λ_{i} t^{α}}$

4.2 Minimal Polynomial Method

Consider that the minimal polynomial of the nth order matrix A is m(λ) = (λ − λ₁)ⁿ^₁ (λ − λ₂)ⁿ^₂ … (λ − λ_k)n_^k, where $\sum_{i = 1}^{k} n_{i} = n$ .

According to the theory of matrix analysis [35], we can express E_α(At^α) as a matrix polynomial of degree n − 1, i.e.,

(15)

E_{α} (A t^{α}) = a_{0} (t) I + a_{1} (t) A + \dots + a_{m - 1} (t) A^{m - 1} .

Eq. (15) holds if and only if E_α(λt^α) and the polynomial ψ(λ, t) = a₀(t)+a₁(t)λ+· · ·+a_m₋₁(t)λ^m⁻¹ are consistent in the spectrum of matrix A; that is,

(16)

\begin{array}{l} {\frac{d^{j} ψ (λ, t)}{d λ^{j}} ∣}_{λ = λ_{i}} = t^{j α} E_{α}^{(j)} (λ_{i} t^{α}), \\ i = 1, 2, \dots k, j = 0, 1, \dots n_{i} - 1. \end{array}

We can obtain a_i(t) by solving Eq. (16).

5. Numerical Examples

5.1 Example

We consider the following SFFDE:

(17)

D_{t}^{a} \tilde{X} (t) = A \tilde{X} (t), A = (\begin{matrix} 2 & - 1 \\ 4 & - 3 \end{matrix}) .

With an initial condition,

\tilde{X} (0) = (\begin{matrix} \underline{x_{1}} (0) \\ \underline{x_{2}} (0) \\ \bar{x_{1}} (0) \\ \bar{x_{2}} (0) \end{matrix}) = (\begin{matrix} r + 1 \\ 2 + r \\ 3 - r \\ 5 - r \end{matrix}),

where λ₁ = 1 and λ₂ = −2 denote the eigenvalues of the matrix A.

By applying the Jordan canonical method, we have A = PJP⁻¹, where

(18)

\begin{array}{l} P = (\begin{matrix} 1 & 1 \\ 1 & 4 \end{matrix}) a n d J = (\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}), \\ E_{α} (A t^{α}) = P E_{α} (A t^{α}) P^{- 1} \\ = P (\begin{matrix} E_{α} (t^{α}) & 0 \\ 0 & E_{α} (- 2 t^{α}) \end{matrix}) P^{- 1} \\ = \frac{1}{3} (\begin{matrix} 4 & - 1 \\ 4 & - 1 \end{matrix}) E_{α} (t^{α}) + \frac{1}{3} (\begin{matrix} - 1 & 1 \\ - 4 & 4 \end{matrix}) E_{α} (- 2 t^{α}) . \end{array}

We apply the minimal polynomial method as follows:

Next, we consider the minimal polynomial of A as m(λ) = (λ − 1)(λ + 2). Further, we consider E_α(At^α) = a₀(t)I₂ + a₁(t)A, where I₂ represents a second-order identity matrix. Thus,

\begin{array}{l} a_{0} (t) + a_{1} (t) = E_{α} (t^{α}), \\ a_{0} (t) - 2 a_{1} (t) = E_{α} (- 2 t^{α}) . \end{array}

By solving the abovementioned system, we obtain

\begin{array}{l} a_{0} (t) = \frac{2}{3} E_{α} (t^{α}) + \frac{1}{3} E_{α} (- 2 t^{α}), \\ a_{1} (t) = \frac{1}{3} E_{α} (t^{α}) - \frac{1}{3} E_{α} (- 2 t^{α}) . \end{array}

In addition, the matrix Mittag-Leffler function has the form

(19)

\begin{array}{l} E_{α} (A t^{α}) = [\frac{2}{3} E_{α} (t^{α}) + \frac{1}{3} E_{α} (- 2 t^{α})] I_{2} + [\frac{1}{3} E_{α} (t^{α}) - \frac{1}{3} E_{α} (- 2 t^{α})] A \\ = \frac{1}{3} (\begin{matrix} 4 & - 1 \\ 4 & - 1 \end{matrix}) E_{α} (t^{α}) + \frac{1}{3} (\begin{matrix} - 1 & 1 \\ - 4 & 4 \end{matrix}) E_{α} (- 2 t^{α}) . \end{array}

The results presented in Eqs. (18) and (19) are the same.

The solution of SFFDE (17) parameterized by the order α is

\tilde{X} (t) = E_{α} (A t^{α}) \tilde{X} (0) .

Then,

\begin{array}{l} \underline{X} (t; r) = (r + \frac{2}{3}) (\begin{matrix} 1 \\ 1 \end{matrix}) E_{α} (t^{α}) + \frac{1}{3} (\begin{matrix} 1 \\ 4 \end{matrix}) E_{α} (- 2 t^{α}), \\ \bar{X} (t; r) = (\frac{7}{3} - r) (\begin{matrix} 1 \\ 1 \end{matrix}) E_{α} (t^{α}) + \frac{2}{3} (\begin{matrix} 1 \\ 4 \end{matrix}) E_{α} (- 2 t^{α}) . \end{array}

The above results are consistent with those obtained by the eigenvalue and eigenvector methods in [23, Eg. 1].

5.2 Example

Consider the following SFFDE:

(20)

D_{t}^{α} \tilde{X} (t) = A \tilde{X} (t), A = (\begin{matrix} 1 & 1 & 1 \\ 2 & 1 & - 1 \\ 0 & - 1 & 1 \end{matrix}) .

With an initial condition,

\tilde{X} (0) = (\begin{matrix} \underline{x_{1}} (0) \\ \underline{x_{2}} (0) \\ \underline{x_{3}} (0) \\ \bar{x_{1}} (0) \\ \bar{x_{2}} (0) \\ \bar{x_{3}} (0) \end{matrix}) = (\begin{matrix} 0.75 + 0.25 r \\ 1.5 + 0.5 r \\ 3.75 + 0.25 r \\ 1.25 - 0.25 r \\ 2.5 - 0.5 r \\ 4.25 - 0.25 r \end{matrix}) .

Subsequently, λ₁ = −1 and λ₂ = λ₃ = 2 denote the eigenvalues of matrix A. By applying the Jordan canonical method, we obtain A = PJP⁻¹, where

(21)

\begin{array}{l} P = (\begin{matrix} - 3 & 0 & 1 \\ 4 & 1 & 0 \\ 2 & - 1 & 1 \end{matrix}), J = (\begin{matrix} - 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{matrix}), \\ E_{α} (A t^{α}) = P E_{α} (A t^{α}) P^{- 1} \\ = P (\begin{matrix} E_{α} (- t^{α}) & 0 & 0 \\ 0 & E_{α} (2 t^{α}) & t^{α} E_{α}^{'} (2 t^{α}) \\ 0 & 0 & E_{α} (2 t^{α}) \end{matrix}) P^{- 1} \\ = \frac{1}{9} (\begin{matrix} 3 & - 3 & - 3 \\ - 4 & 4 & 4 \\ - 2 & 2 & 2 \end{matrix}) E_{α} (- t^{α}) + \frac{1}{9} (\begin{matrix} 6 & 3 & 3 \\ 4 & 5 & - 4 \\ 2 & - 2 & 7 \end{matrix}) E_{α} (2 t^{α}) + \frac{1}{9} (\begin{matrix} 0 & 0 & 0 \\ 6 & 3 & 3 \\ - 6 & - 3 & - 3 \end{matrix}) t^{α} E_{α}^{'} (2 t^{α}) . \end{array}

Further, we apply the minimal polynomial method as follows:

The minimal polynomial of A is m(λ) = (λ + 1)(λ − 2)² and E_α(At^α) = a₀(t)I₃ + a₁(t)A + a₂(t)A². Thus,

\begin{array}{l} a_{0} (t) - a_{1} (t) + a_{2} (t) = E_{α} (- t^{α}), \\ a_{0} (t) + 2 a_{1} (t) + 4 a_{2} (t) = E_{α} (2 t^{α}), \\ a_{1} (t) + 4 a_{2} (t) = t^{α} E_{α}^{'} (2 t^{α}) . \end{array}

We now solve the system to obtain

\begin{array}{l} a_{0} (t) = \frac{4}{9} E_{α} (- t^{α}) + \frac{5}{9} E_{α} (2 t^{α}) - \frac{6}{9} t^{α} E_{α}^{'} (2 t^{α}), \\ a_{1} (t) = - \frac{4}{9} E_{α} (- t^{α}) + \frac{4}{9} E_{α} (2 t^{α}) - \frac{3}{9} t^{α} E_{α}^{'} (2 t^{α}), \\ a_{2} (t) = \frac{1}{9} E_{α} (- t^{α}) - \frac{1}{9} E_{α} (2 t^{α}) + \frac{3}{9} t^{α} E_{α}^{'} (2 t^{α}) . \end{array}

Then, the matrix Mittag-Leffler function assumes the form

(22)

\begin{array}{l} E_{α} (A t^{α}) = & [\frac{4}{9} E_{α} (- t^{α}) + \frac{5}{9} E_{α} (2 t^{α}) - \frac{6}{9} t^{α} E_{α}^{'} (2 t^{α})] I_{3} \\ + [- \frac{4}{9} E_{α} (- t^{α}) + \frac{4}{9} E_{α} (2 t^{α}) - \frac{3}{9} t^{α} E_{α}^{'} (2 t^{α})] A \\ + [\frac{1}{9} E_{α} (- t^{α}) - \frac{1}{9} E_{α} (2 t^{α}) + \frac{3}{9} t^{α} E_{α}^{'} (2 t^{α})] A^{2} \\ = & \frac{1}{9} (\begin{matrix} 3 & - 3 & - 3 \\ - 4 & 4 & 4 \\ 2 & 2 & 2 \end{matrix}) E_{α} (- t^{α}) + \frac{1}{9} (\begin{matrix} 6 & 3 & 3 \\ 4 & 5 & - 4 \\ 2 & - 2 & 7 \end{matrix}) E_{α} (2 t^{α}) \\ + \frac{1}{9} (\begin{matrix} 0 & 0 & 0 \\ 6 & 3 & 3 \\ - 6 & - 3 & - 3 \end{matrix}) t^{α} E_{α}^{'} (2 t^{α}) . \end{array}

The results presented in Eqs. (21) and (22) are the same. The solution of SFFDE (20) parameterized by the order α is

\tilde{X} (t) = E_{α} (A t^{α}) \tilde{X} (0) .

Then,

\begin{array}{l} \underline{X} (t; r) = & \frac{1}{9} (\begin{matrix} - 13.5 - 1.5 r \\ 18 + 2 r \\ 9 + r \end{matrix}) E_{α} (- t^{α}) \\ + \frac{1}{9} (\begin{matrix} 20.25 + 3.75 r \\ - 4.5 + 2.5 r \\ 24.75 + 1.25 r \end{matrix}) E_{α} (2 t^{α}) \\ + \frac{1}{9} (\begin{matrix} 0 \\ 20.25 + 3.75 r \\ - 20.25 - 3.75 r \end{matrix}) t^{α} E_{α}^{'} (2 t^{α}), \\ \bar{X} (t; r) = & \frac{1}{9} (\begin{matrix} - 16.5 + 1.5 r \\ 22 - 2 r \\ 11 - r \end{matrix}) E_{α} (- t^{α}) \\ + \frac{1}{9} (\begin{matrix} 27.75 - 3.75 r \\ 0.5 - 2.5 r \\ 27.25 - 1.25 r \end{matrix}) E_{α} (2 t^{α}) \\ + \frac{1}{9} (\begin{matrix} 0 \\ 27.75 - 3.75 r \\ - 27.75 + 3.75 r \end{matrix}) t^{α} E_{α}^{'} (2 t^{α}) . \end{array}

The solutions presented above can be rewritten in the following form:

\begin{array}{l} \underline{X} (t; r) = & \frac{1}{9} (4.5 + 0.5 r) (\begin{matrix} - 3 \\ 4 \\ 2 \end{matrix}) E_{α} (- t^{α}) \\ + \frac{1}{9} (- 4.5 + 2.5 r) (\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}) E_{α} (2 t^{α}) \\ + \frac{1}{9} (20.25 + 3.75 r) [(\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}) t^{α} E_{α}^{'} (2 t^{α}) + (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) E_{α} (2 t^{α})], \\ \bar{X} (t; r) = & \frac{1}{9} (5.5 - 0.5 r) (\begin{matrix} - 3 \\ 4 \\ 2 \end{matrix}) E_{α} (- t^{α}) \\ + \frac{1}{9} (0.5 - 2.5 r) (\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}) E_{α} (2 t^{α}) \\ + \frac{1}{9} (27.75 - 3.75 r) [(\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}) t^{α} E_{α}^{'} (2 t^{α}) + (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}) E_{α} (2 t^{α})] . \end{array}

The above results are consistent with those obtained by the eigenvalue and eigenvector methods in [23, Eg. 2].

6. Conclusion

Abstract
1. Introduction
2. Basic Concepts
3. Solution for a System of Fuzzy FDEs
4. Calculation of the Matrix Mittag-Leffler Function
4.2 Minimal Polynomial Method
5. Numerical Examples
6. Conclusion

Fig 1.

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Figure 1.

Solutions x̃₁(t) and x̃₂(t) of Example 1 for different values of (a) α = 0.6, (b) α = 0.8, (c) α = 1.0 at t = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 144-154https://doi.org/10.5391/IJFIS.2022.22.2.144

Fig 2.

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Figure 2.

Solutions x̃₁(t), x̃₂(t), and x̃₃(t) of Example 2 for different values of (a) α = 0.6, (b) α = 0.8, (c) α = 1.0 at t = 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 144-154https://doi.org/10.5391/IJFIS.2022.22.2.144

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

Abstract

1. Introduction

2. Basic Concepts

Definition 2.1 [25]

Definition 2.2 [25,26]

Definition 2.3 [27]

Definition 2.4 [28]

Definition 2.5 [29]

Definition 2.6 [30,31]

Definition 2.7 [31,32]

Definition 2.8

3. Solution for a System of Fuzzy FDEs

Theorem 3.1

Theorem 3.2

4. Calculation of the Matrix Mittag-Leffler Function

4.1 Matrix of the Jordan Canonical Method

4.2 Minimal Polynomial Method

5. Numerical Examples

5.1 Example

5.2 Example

6. Conclusion

Conflict of Interest

Figures

References

Biographies

Article

Original Article

Solutions of a System of Fuzzy Fractional Differential in Terms of the Matrix Mittag-Leffler Functions

Abstract

1. Introduction

2. Basic Concepts

Definition 2.1 [25]

Definition 2.2 [25,26]

Definition 2.3 [27]

Definition 2.4 [28]

Definition 2.5 [29]

Definition 2.6 [30,31]

Definition 2.7 [31,32]

Definition 2.8

3. Solution for a System of Fuzzy FDEs

Theorem 3.1

Theorem 3.2

4. Calculation of the Matrix Mittag-Leffler Function

4.1 Matrix of the Jordan Canonical Method

4.2 Minimal Polynomial Method

5. Numerical Examples

5.1 Example

5.2 Example

6. Conclusion

Fig 1.

Fig 2.

References

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