Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense
Abstract
:1. Introduction
2. Preliminaries
- is upper semi-continuous, i.e., , .
- is convex, i.e., for each , and , we have .
- is normal, i.e., there is at least one pointsuch that
- is compact set.
- (i)
- is a bounded non-decreasing function.
- (ii)
- is a bounded non-increasing function.
- (iii)
- .
- (iv)
- For each,and.
- (v)
- and.
- (i)
- The H-differences,exist for each sufficiently small, and.
- (ii)
- The H-differences,exist, for each sufficiently small, and.
- (i)
- For each there exists such that the H-differences: and exists for allζ∈ [0,δ).
- (ii)
- For eachandthere exists a constantsuch thatandfor all. Then, the set of functionsis-th conformable differentiable and its derivative is, wherefor each
- (i)
- If is -fuzzy conformable differentiable, then and are -th conformable differentiable functions on and .
- (ii)
- If is -fuzzy conformable differentiable, then and are -th conformable differentiable functions on and .
3. Fuzzy Conformable Fractional Initial Value Problems
- (1)
- If is -fuzzy conformable differentiable, then the corresponding crisp system of the IVPs (1) and (2) will be written in the form of the following:
- (2)
- If is -fuzzy conformable differentiable, then the corresponding crisp system of IVPs (1) and (2) will be written in the form of the following:The formulation of the fuzzy fractional IVPs (1) and (2) along with Theorem 2.3 show us how to deal with numerical solutions of fuzzy fractional IVPs. The original fuzzy fractional IVPs can be converted into a crisp system of fractional IVPs equivalently. This indicates that no need to rewrite the numerical methods for the crisp systems of the fractional IVPs in the fuzzy setting, but, instead, we may use the numerical methods directly on the obtained crisp systems.
4. Primary Principle of Residual Power Series Approach
- ▪
- and for each .
- ▪
- and for each .
- ▪
- and for .
Algorithm 1. To deduce the approximate solutions of (3) in detail, one can perform the following manner by one of the known software packages like MATHEMATICA 12. |
Step I: Write the system (3) in the form
|
Step II: Suppose that the solutions of the system (3) about the initial point have the fractional PS expansion forms
|
Step III: Set and , then the -th fractional PS approximate solutions and of can be written respectively as
|
Step IV: Define the -th fractional residual functions and such that
|
Step V: Substitute and in and so that
|
Step VI: Consider , in Step V, then solve and at for and . Therefore, the first fractional PS approximate solutions and will be obtained. |
Step VII: For in Step V, apply the operator -th on both sides of the resulting fractional equations such that and . Then, by solving and , and can be obtained. |
Step VIII: Write the forms of the obtained coefficients and in terms of -th fractional PS expansions and and repeat the above steps to reach a closed-form in terms of infinite series as in Step II. Elsewhere, the solution obtained will be representing the -th fractional PS approximate solutions of the crisp system (3). |
5. Applications and Numerical Simulations
, | , |
, | , |
, | , |
, | , |
, | , |
, | , |
, | , |
, | , |
, | , |
, | , |
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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RPSM | RKHSM | RPSM | RKHSM | ||
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Bataineh, M.; Alaroud, M.; Al-Omari, S.; Agarwal, P. Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense. Entropy 2021, 23, 1646. https://doi.org/10.3390/e23121646
Bataineh M, Alaroud M, Al-Omari S, Agarwal P. Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense. Entropy. 2021; 23(12):1646. https://doi.org/10.3390/e23121646
Chicago/Turabian StyleBataineh, Malik, Mohammad Alaroud, Shrideh Al-Omari, and Praveen Agarwal. 2021. "Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense" Entropy 23, no. 12: 1646. https://doi.org/10.3390/e23121646