[Paper Review] A Note on "Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations"

Robab Alikhani, F. Bahrami|arXiv (Cornell University)|Dec 10, 2014

Fuzzy Systems and Optimization3 references8 citations

TL;DR

This paper validates the exact solutions in two examples from a prior study on fuzzy fractional integrodifferential equations by rigorously proving their correctness using α-level sets and the L-U representation of fuzzy numbers. It refutes claims by another study that the solutions were invalid due to incorrect use of x − x = 0 for non-real fuzzy numbers, demonstrating instead that the solutions satisfy the equations through interval arithmetic on α-cuts, confirming their validity within the framework of upper and lower solutions.

ABSTRACT

The authors in \cite{alikhani} have given two examples to illustrate their results in which they have been eliminated the technical details. However, the authors in \cite{salahshur} claimed that the examples are incorrect. In fact they conjectured that the authors in \cite{alikhani} have employed the incorrect statement $x-x=0$ for $x\in {\mathbb R}_{\mathcal F}\setminus \mathbb R$ to construct the examples. Here we intend to observe that the basic method used in \cite{alikhani} to prove the validity of the examples is the well-known L-U representation of a fuzzy-number valued function. In this sense, we will make use of the $α$-level sets and show that the examples are correct.

Motivation & Objective

• To resolve a dispute over the correctness of exact solutions in two examples from a prior study on fuzzy fractional integrodifferential equations (FFIDEs).
• To demonstrate that the solutions are mathematically valid despite claims of error by another study.
• To clarify that the use of x − x = 0 for non-real fuzzy numbers was not employed in the original proofs.
• To show that the solutions lie within the bounds defined by upper and lower solutions, as required by the theoretical framework.
• To establish that the method of proof based on α-level sets is sufficient and correct for verifying solutions in fuzzy fractional calculus.

Proposed method

• The paper uses α-level sets to represent fuzzy numbers, expressing each fuzzy number c as [c]α = [clα, crα] for α ∈ [0,1].
• It applies the L-U representation of fuzzy-number-valued functions to analyze the behavior of solutions across α-cuts.
• The validity of the solutions is verified by checking the equality of interval-valued expressions derived from the fractional integrodifferential equation on each α-level.
• It employs interval arithmetic to compute the left-hand and right-hand sides of the equations in the examples, showing they match for all α ∈ [0,1].
• The proof relies on the properties of fuzzy arithmetic, particularly the distributive and scalar multiplication rules from Lemma 2.2, avoiding invalid operations like x − x = 0 for non-real fuzzy numbers.
• It references Theorem 4.5–4.6 from the original study to confirm that solutions need only lie between upper and lower solutions for some, not all, solutions.

Experimental results

Research questions

• RQ1 Are the exact solutions presented in Examples 1 and 2 of [1] mathematically valid despite claims of error?
• RQ2 Did the original authors incorrectly assume x − x = 0 for fuzzy numbers not in ℝ?
• RQ3 Can the correctness of the solutions be established using α-level sets instead of fuzzy arithmetic identities?
• RQ4 Is it necessary for all solutions to lie strictly between upper and lower solutions in the context of fuzzy fractional equations?
• RQ5 Do the solutions u(t) = c and u(t) = c + ct^{q-1} satisfy the initial value problems under the given definitions of fuzzy fractional derivatives?

Key findings

• The solution u(t) = c is valid for Example 2.4, as the α-level sets of both sides of the equation match after applying interval arithmetic.
• The solution u(t) = c + ct^{q-1} is valid for Example 2.5, with equality confirmed on each α-cut using interval operations.
• The claim in [4] that x − x = 0 was used in the proofs is incorrect; the paper shows that the proof relies solely on α-level set arithmetic, not on fuzzy subtraction.
• The solution u(t) = c satisfies the initial condition lim_{t→0+} t^{1−q}u(t) = ˆ0, as required by Definition 2.3.
• The solution u(t) = c + ct^{q−1} lies within the bounds defined by the lower solution ct^{q−1} and upper solution 10ct^{q−1} when c ≥ ˆ0, consistent with Theorem 4.5 in [1].
• The paper confirms that not all solutions need to lie between upper and lower solutions—only some solutions are required to do so, which resolves the misunderstanding in [4].