Correction: Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag

Gamal A. Mosa; Mohamed A. Abdou; Ahmed S. Rahby; Gamal A. Mosa; Mohamed A. Abdou; Ahmed S. Rahby

doi:10.3934/math.2022016

AIMS Mathematics

2022, Volume 7, Issue 1: 258-259. doi: 10.3934/math.2022016

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Correction

Correction: Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag

1.
Department of Mathematics, Faculty of Science, Benha University, Egypt
2.
Department of Mathematics, Faculty of Education, Alexandria University, Egypt
Correction of: AIMS Mathematics 6: 8525-8543

Received: 27 September 2021 Accepted: 27 September 2021 Published: 11 October 2021

Citation: Gamal A. Mosa, Mohamed A. Abdou, Ahmed S. Rahby. Correction: Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag[J]. AIMS Mathematics, 2022, 7(1): 258-259. doi: 10.3934/math.2022016

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A correction on

Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag

by G. A. Mosa, M. A. Abdou and A. S. Rahby. AIMS Mathematics, 2021, 6(8): 8525–8543.

DOI: 10.3934/math.2021495

These errata give the following correct statements for the corresponding statements on the cited page of our published article^[1].

In page 8528, in the first line we should replace "plan" with "plane" and "Eq (2.3)" by "Eq (2.2)". In addition, correcting of (2.4) as given below in (0.1).

$\begin{equation} \begin{split} |\lambda| < \frac{|\mu|(2q-T^2)}{ A_1 D_1 \left(2q B_2T+B_3T^2\right) }. \end{split} \end{equation}$

(0.1)

In page 8529, correcting of $\eta_1\!\! = \!\!\frac{T}{q}\!+\!|\frac{\lambda}{q\mu}| A_1 D_1 (qB_2\!+\!TB_3)\!\!<\!\!1$ in (2.10) is $\eta_1\!\! = \!\!\frac{T^2}{2q}+\frac{|\lambda|}{q |\mu|} A_1 D_1\! \left(q B_2T\!\!+\!\!B_3\frac{T^2}{2}\right)\!<\!\!1.$ Therefore, correcting (2.11) as given in (0.1).

In page 8530, correcting of (2.14) is

$\begin{equation} \begin{split} \overline{W}\phi& = q\mu H(x,t)+ W \phi \ \text{and} \ \ \overline{W}\phi = q \mu \phi, \end{split} \end{equation}$

(0.2)

where

$W\phi = -W_1\phi+ W_2\phi+W_3\phi,\ \ W_1\phi = \mu\int_0^t \phi(x,z)dz,$

$\!\!\!\!W_2\phi = \!\! \lambda \!\!\int_{0}^{q}\!\!\!\int_{a}^{b}\!\!\!\!\!\Theta(t,\tau)K(x,y) G(y, \tau, \phi(y,\tau)) dy d \tau,$

and

$\ W_3\phi = \!\! \lambda\!\!\int_{q}^{ t+q}\!\! \!\int_{a}^{b}\!\!\! \Psi(t,\tau) K(x,y) G(y, \tau, \phi(y,\tau)) dy d \tau.$

Moreover, (2.15) becomes

$\begin{equation} \begin{split} \|W\phi\|&\leq\|\mu \int_0^t \phi(x,z)dz \|+\|\lambda \int_0^{q} \int_{a}^{b}\ \Theta(t,\tau)\ K(x,y) G(y, \tau, \phi(y,\tau)) dy \ d \tau\|\\ &\quad+\|\lambda \int_q^{t+q} \int_{a}^{b}\ \Psi(t,\tau) K(x,y) G(y, \tau, \phi(y,\tau)) dy \ d \tau\|. \end{split} \end{equation}$

(0.3)

In addition, correcting of $\eta_2\!\! = \!\!\frac{T}{q}\! +\!|\frac{\lambda}{q} |(q B_2\!+\!T B_3)A_1 D_2\!<\!\!1$ in (2.16) is $\eta_2\! = \!|\mu|\frac{T^2}{2}\!+\!|\lambda|A_1D_2 (q TB_2\!+\!B_3\frac{T^2}{2})\!<\!\!1$ and correcting of the last inequality in Section $2.2.1$ is $|\lambda|<\frac{2-|\mu|T^2}{\left( 2q TB_2+B_3T^2\right)A_1 D_2 }.$

Also, (2.17) becomes

$\begin{equation} \begin{split} & \| \overline{W}\phi_1-\overline{W}\phi_2\| = \| W\phi_1-W\phi_2\|\leq \eta_3 \| \phi_1- \phi_2\| , \\ &\quad \eta_3 = |\mu|\frac{T^2}{2}+|\lambda|A_1D_1 (q TB_2+B_3\frac{T^2}{2}) < 1. \end{split} \end{equation}$

(0.4)

In page 8531, correcting of the first inequality is $|\lambda|<\frac{2-|\mu|T^2}{\left( 2q TB_2+B_3T^2\right)A_1 D_1 }.$

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	G. A. Mosa, M. A. Abdou, A. S. Rahby, Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag, AIMS Math., 6 (2021), 8525–8543. doi: 10.3934/math.2021495. doi: 10.3934/math.2021495

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沈阳化工大学材料科学与工程学院沈阳 110142

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