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Abstract

This study presents an innovative application of Feedforward Neural Networks ‘FNNs’ to solve Variable-Order Fractional Partial Differential Equations ‘VO-FPDEs’ with time delays. Utilizing the Caputo definition, the variable-order fractional derivatives are approximated in terms of integer-order derivatives. The problem is reformulated as a system of partial differential equations with delay terms, which is then addressed using ‘FNNs’ to achieve explicit approximate solutions. Comprehensive error and convergence analyses validate the method’s precision and reliability. The effectiveness of the proposed approach is highlighted through numerical examples, with graphical and tabular representations showcasing minimal absolute errors and robust convergence. These results demonstrate the proposed method’s efficiency and simplicity, establishing it as a powerful tool for addressing complex fractional delay problems.

Reason for Retraction

This article has been retracted at the request of the Editorial Office, following an internal investigation conducted in accordance with the Committee on Publication Ethics (COPE) Retraction Guidelines.

The investigation identified serious concerns affecting the integrity and reliability of the published work. Specifically, one or more of the following issues were confirmed:

  1. Undisclosed use of computer-generated text and/or data, in which substantial portions of the content were produced using algorithmic or artificial intelligence–based tools without transparent disclosure, contrary to the journal's authorship and transparency policies.

  2. Compromised peer-review process, indicating irregularities that undermine the validity, independence or authenticity of the review procedure.

  3. Inappropriate or misleading citations, including references that are irrelevant, improperly used, or appear to artificially inflate citation metrics, thereby distorting the scholarly record.

  4. Authorship-related concerns, including the addition of new author(s) at a later stage of the publication process without adequate justification, documentation, or transparent disclosure, raising unresolved questions regarding author contributions, responsibility, and compliance with the journal's authorship criteria.

The Editorial Office determined that these issues significantly compromise the scientific integrity of the article, and that correction alone would be insufficient to address the concerns. Retraction was therefore deemed necessary to maintain the accuracy and trustworthiness of the scholarly record.

The authors were informed of the findings and the retraction decision. While the authors do not respond to this retraction, the journal has proceeded with the retraction in line with COPE guidance, which permits retraction without author consent when editorial integrity is at risk.

This retraction is issued to alert readers that the findings and conclusions of the article should not be relied upon. The original article will remain accessible for the sake of the scholarly record, but it will be clearly marked as retracted.

Apologies are offered to readers of the journal that this was not detected during the submission process.

Please see the Retraction Notice available at: https://ijcsm.researchcommons.org/ijcsm/vol6/iss3/12

DOI: https://doi.org/10.52866/2788-7421.1376

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