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Abstract

This study aims to present the modified Chebyshev wavelet collocation method (CWCM) to investigate and obtain the numerical approximation of $CO_2$ emissions from the energy sector utilizing the fractional mathematical model. The need for energy rises as the population grows. Burning fossil fuels produces a significant portion of the world's energy, which raises the atmospheric concentration of $CO_2$ and causes global warming. The combination of mathematical modeling studies and numerical simulations allows us to understand the $CO_2$ emissions from the energy sector. Our objective is to build an operational matrix of integration (OMI) based on Chebyshev wavelets and use it to the numerical solution of the $CO_2$ emanation from the energy sector model. Furthermore, these OMI are used to simplify fractional differential equations into an algebraic equation system. Lastly, graphic representations are used to visually present the results and provide empirical support for our theoretical conclusions. Because the CWCM method computes only a few terms and avoids rounding data, it produces correct answers for highly nonlinear situations. We compared the outcomes of the RK4 technique, the ND solver, and the devised method. The numerical results show that CWCM solves the emanation of $CO_2$ from the energy sector with remarkable accuracy and efficiency in both mathematical and environment view. Software called Mathematica has been used for numerical implementation and calculations.

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This article has been retracted at the request of the Editorial Office, following an internal investigation conducted in accordance with the Committee on Publication Ethics (COPE) Retraction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/iss4/2

DOI: https://doi.org/10.52866/2788-7421.1355

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