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Piecewise implicit coupled system under ABC fractional differential equations with variable order

• Saleh S. Redhwan 1 ,  ,  , 
• Maoan Han 1 , 
• Mohammed A. Almalahi 2 , 
• Maryam Ahmed Alyami 3 , 
• Mona Alsulami 3 , 
• Najla Alghamdi 3
• 1. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, China
• 2. Department of Mathematics, Hajjah University, Hajjah, Yemen
• 3. Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah, Saudi Arabia
• Received: 23 February 2024 Revised: 13 April 2024 Accepted: 16 April 2024 Published: 28 April 2024

MSC : 34A08, 34A12, 47H10, 97M70

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This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research.

• ABC-fractional differential equation ,
• nonlocal conditions ,
• fixed point theorem

Citation: Saleh S. Redhwan, Maoan Han, Mohammed A. Almalahi, Maryam Ahmed Alyami, Mona Alsulami, Najla Alghamdi. Piecewise implicit coupled system under ABC fractional differential equations with variable order[J]. AIMS Mathematics, 2024, 9(6): 15303-15324. doi: 10.3934/math.2024743

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Computer Science > Computation and Language

Title: mathprompter: mathematical reasoning using large language models.

Abstract: Large Language Models (LLMs) have limited performance when solving arithmetic reasoning tasks and often provide incorrect answers. Unlike natural language understanding, math problems typically have a single correct answer, making the task of generating accurate solutions more challenging for LLMs. To the best of our knowledge, we are not aware of any LLMs that indicate their level of confidence in their responses which fuels a trust deficit in these models impeding their adoption. To address this deficiency, we propose `MathPrompter', a technique that improves performance of LLMs on arithmetic problems along with increased reliance in the predictions. MathPrompter uses the Zero-shot chain-of-thought prompting technique to generate multiple Algebraic expressions or Python functions to solve the same math problem in different ways and thereby raise the confidence level in the output results. This is in contrast to other prompt based CoT methods, where there is no check on the validity of the intermediate steps followed. Our technique improves over state-of-the-art on the MultiArith dataset ($78.7\%\rightarrow92.5\%$) evaluated using 175B parameter GPT-based LLM.

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Column generation based solution for bi-objective gate assignment problems

• Original Article
• Published: 29 April 2024

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• Gülesin Sena Daş 1 , 2 &
• Fatma Gzara 3  

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In this paper, we present a column generation-based algorithm for the bi-objective gate assignment problem (GAP) to generate gate schedules that minimize squared slack time at the gates while satisfying passenger expectations by minimizing their walking distance. While most of the literature focuses on heuristic or metaheuristic solutions for the bi-objective GAP, we propose flow-based and column-based models that lead to exact or near optimal solution approaches. The developed algorithm calculates a set of solutions to approximate the Pareto front. The algorithm is applied to the over-constrained GAP where gates are a limited resource and it is not possible to serve every flight using a gate. Our test cases are based on real data from an international airport and include various instances with flight-to-gate ratios between 23.9 and 34.7. Numerical results reveal that a set of solutions representing a compromise between the passenger-oriented and robustness-oriented objectives may be obtained with a tight optimality gap and within reasonable computational time even for these difficult problems.

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The first author is supported by Scientific and Technological Research Council of Turkey (TUBITAK) (Grant No. 1059B191700275) 2219 Post Doctoral Research Fellowship Program during her research at WAnOpt Lab, University of Waterloo.

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Gülesin Sena Daş

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Daş, G.S., Gzara, F. Column generation based solution for bi-objective gate assignment problems. Math Meth Oper Res (2024). https://doi.org/10.1007/s00186-024-00856-1

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Received : 26 June 2022

Revised : 22 February 2024

Accepted : 03 March 2024

Published : 29 April 2024

DOI : https://doi.org/10.1007/s00186-024-00856-1

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