Correction: Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

A correction on

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

by A. H. Tedjani, A. Z. Amin, Abdel-Haleem Abdel-Aty, M. A. Abdelkawy, and Mona Mahmoud. AIMS Mathematics, 2024, 9(4): 7973–8000. DOI: 10.3934/math.2024388

The author would like to make the following corrections to the published paper ^[1].

● In the abstract of our paper ^[1], we stated: "In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem". Acutully, this statement requires further clarification. Specifically, we address the challenge posed by non-smooth solutions, which can significantly degrade the performance of numerical schemes, particularly their order of convergence. To overcome this limitation, we employed fractional-order Legendre functions, denoted as $\mathcal{L}_{\varepsilon}(\lambda^{\gamma})$, in our numerical approach. This methodology was applied specifically in Examples 6 and 7, as mentioned in the conclusion section of ^[1].

● The corrected form of Eqs (1.1) and (1.2) are

with the initial conditions;

where $D^{\alpha_{1}}$ denotes the fractional derivative of order $\alpha_{1}$, and $0 < \alpha_{1} < 2$ (This modification must be applied consistently across the paper).

● Using the following transformations $t = 2\lambda-1$, $s = 2\varrho-1$, $\mathcal{Y}(2\varrho-1) = \mathcal{Z}(\varrho)$, $\phi(2\varrho-1) = \phi(\varrho)$, $2G(\mathcal{Y}(2\lambda-1) = F(\mathcal{Z}(\lambda))$, and $\eta(2\varrho-1, 2\lambda-1) = \sigma(\varrho, \lambda)$ we obtain, Eq (3.12) in ^[1].

● The initial conditions (3.12) must change to be

● Equation (3.17) must change to be

where

● The system of $(\nu_1 + 1)$ algebraic equations derived in Eq (3.23) constitutes a nonlinear system.

● Lemma 3. Consider $e(x) = \mathcal{Z}(\varrho)-\mathcal{Z} N(\varrho)$ to represent the error function of the solution. The subsequent inequality is applicable in this context:

where

$B_{1} = I_{\varrho, N}D^{\alpha_{1}}\mathcal{Z}(\varrho)-D^{\alpha_{1}}\mathcal{Z}(\varrho)$

$B_{2} = I_{\varrho, N}\int_{-1}^{1}(I-I_{\lambda, N})\bigg[\sigma(\varrho, \lambda)F(\mathcal{Z}(\lambda))\bigg]d\lambda$

$B_{3} = I_{\varrho, N}\int\limits_{-1}^{1}I_{\lambda, N}\bigg[\sigma(\varrho, \lambda)F(\mathcal{Z}(\lambda))-\sigma(\varrho, \lambda)F(\mathcal{Z}_{N}(\lambda))\bigg]d\lambda$.

Proof. By using the Caputo definition, we write the equation of non-FFIDEs as follows:

and when utilizing the approximate solution we have,

Subtracting (0.7) from (0.6) yields

hence

The desired result can be obtained directly from the above.

● In Theorem 1, Section 4.1, we compute $B_1$, instead of Eq (4.11), by using Lemma 3, Lemma (3-3) in ^[2], as

Accordingly, Eq (4.8) must be

● In Eq (4.17), which $L$ is Lipschitz condition, and $Max|\sigma(\varrho, \lambda)|\leq M$ and $L < 1/M$.

● The revised version of Figure 8 in ^[1] is included in Figure 1.

Figure 1.  The approximate solutions for various values of $\alpha_{1}$.

DownLoad: Full-Size Img PowerPoint

Theorem 1. Let $I_{N}\mathcal{Z}(\varrho)$ be the spectral approximate and let $\mathcal{Z}(\varrho)$ be the exact solution of the equation of non-FFIDEs and, F satisfies the Lipschitz condition with respect to its third argument with the Lipschitz constant $L < \frac{1}{M}$ and $Max|\sigma(\varrho, \lambda)|\leq M$.

● In Example 6, while the solution is not smooth, then the order of convergence of the numerical scheme may deteriorate. However, this can be prevented by using fractional order Legendre functions $\mathcal{L}_{\varepsilon}(\lambda^{\gamma})$. Then, in Figures 13 and 14, $\gamma = \frac{1}{2}$ and $\nu_1 = 8$. Also, we used fractional order Legendre functions $\mathcal{L}_{\varepsilon}(\lambda^{\gamma})$ in Example 7 and then $\alpha_1$ in Table 6 must be $\gamma$.

The changes have no material impact on the conclusion of this article. The original manuscript will be updated ^[1]. We apologize for any inconvenience caused to our readers by the changes.

Conflict of interest

The authors declare there is no conflict of interest.