Corrigendum to “Phase-lag mixed integral equation of a generalized symmetric potential kernel and its physical meanings in (3+1) dimensions” [Heliyon Volume 11, Issue 3, February 2025, Article e42316]

In this article, the author accidently included Figure 5 twice, appearing in place of Figure 4. Additionally reference 22 was duplicated, appearing also as reference 27 in section "9. Linear algebraic system". The authors confirm that the figure and citation duplication were introduced in the article due to an inadvertent error. They have provided the correct Figure 4, removed reference 27, and replaced the in-text citation of reference 27 with reference 22.

The original Figure 4 can be found below:

[Original Figure 4]

The correct version of Figure 4 should be as below

[New Figure 4]

The original references can be found below:

[22] M. Basseem, Degenerate method in mixed nonlinear three dimensions integral equation, Alex. Eng. J. 58 (2019) 387–392.

[27] M. Basseem, Degenerate method in mixed nonlinear three dimensions integral equation, Alex. Eng. J. 58 (2019) 387–392.

The original section "9. Linear algebraic system" can be found below:

"...Using the degenerate kernel method (Basseem [27]), with some algebraic relations of Jacobi polynomials, the following system can be obtained:..."

The reference list should be corrected so that reference 27 is no longer considered in the article, and it is removed from where it was originally cited and replaced with reference 22.

The updated section "9. Linear algebraic system" can be found below:

"...Using the degenerate kernel method (Basseem [22]), with some algebraic relations of Jacobi polynomials, the following system can be obtained:..."

The authors apologize for the errors.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Figure 4. Potential kernel function.

Figure 5. Potential kernel function.

Figures 4 and 5 display 3D plots of the potential kernel function $W_{50}^{0.5}\left(u,v;t=0.1\right))$ under different parameterizations. These plots feature multiple sharp peaks, indicating regions of high intensity in the system's response, such as stress or energy density. The variations in peak magnitude and distribution across different parameter settings highlight the sensitivity of the material's behavior to changes in these parameters. As the parameter $m$ increases from $5$ to $50$, the plots reveal a significant change in the pattern and intensity of the peaks in the potential kernel function. For $m=5$, the peaks are moderately sharp and distributed evenly, indicating a balanced distribution of stress or potential energy across the material. In contrast, with $m=50$, the peaks become more numerous and significantly higher, indicating a greater concentration of energy or stress in specific areas.