Correction: Gao et al. Applications of the Separation of Variables Method and Duhamel’s Principle to Instantaneously Released Point-Source Solute Model in Water Environmental Flow. Sustainability 2024, 16, 6912

The authors would like to make the following corrections to the published paper [1]. The changes are as follows:

(1)

Replacing the sentence in “Section 3.1. Zeroth-Order Concentration Moment Solution”:

Considering the boundary condition of c₂ = 0, m₀ is a constant. It does not meet the requirements, so <i> can be ruled out.

with

Considering the boundary condition, we have c₂ = 0, and c₁ and c₃ are both non-zero (otherwise it would lead to a zero solution). Therefore, for λ = 0, we obtain a constant function (non-zero), which will be utilized in <iii> (for λ < 0).

(2)

Replacing the sentence in “Section 3.1. Zeroth-Order Concentration Moment Solution”:

We have ${\varphi }_2\left(\zeta \right)=c_1{e}^{\sqrt{\lambda }\zeta }+c_2{e}^{-\sqrt{\lambda }\zeta }$, and when considering the initial-boundary conditions, it becomes evident that $c_1=c_2=0$, which does not meet the requirements.

with

We have ${\varphi }_2\left(\zeta \right)=c_1{e}^{\sqrt{\lambda }\zeta }+c_2{e}^{-\sqrt{\lambda }\zeta }$, and when considering the initial boundary conditions, it becomes evident that $c_1=c_2=0$, which leads to a zero solution.

(3)

The authors would like to replace Equation (31):

$$\begin{array}{l}{\varphi }_2\left(\zeta \right)=c_n\cos \left(n\pi \zeta \right),\\ \lambda =-{\left(n\pi \right)}^2,n=0,1,2,\dots \end{array}$$

(31)

with

$${\varphi }_2\left(\zeta \right)=A\cdot \cos \left(\sqrt{-\lambda }\cdot \zeta \right)+B\cdot \sin \left(\sqrt{-\lambda }\cdot \zeta \right)$$

Using the boundary conditions, we can obtain B = 0, then,

$${\varphi }_2\left(\zeta \right)=A\cdot \cos \left(\sqrt{-\lambda }\cdot \zeta \right)$$

and

$$\lambda =-{\left(n\pi \right)}^2,n=1,2,3,\dots$$

Thus, for ${\varphi }_2$,

$$\begin{array}{l}{\varphi }_2\left(\zeta \right)=A_n\cdot \cos \left(n\pi \zeta \right)\\ n=1,2,3,\dots \end{array}$$

(31)

(4)

The authors would like to replace Equation (33):

$${\varphi }_1\left(t^{\ast }\right)=a_n{e}^{-n^2{\pi }^2{t}^{\ast }}$$

(33)

with

$${\varphi }_1\left(t^{\ast }\right)=C\cdot e^{-\lambda t^{\ast }}=C_n\cdot e^{-n^2{\pi }^2{t}^{\ast }}$$

(33)

Thus, we obtain a family of functions:

$$\left\{c_n\cdot \cos \left(n\pi \zeta \right)\cdot e^{-n^2{\pi }^2{t}^{\ast }}\right\},n=1,2,3,\cdots$$

Here, $c_n=A_n\cdot C_n$.

(5)

Replacing the sentence in “Section 3.1. Zeroth-Order Concentration Moment Solution”:

According to the value of the indicator n, we can create a linear superposition. The analytical expression of $m_0$ is given as follows:

with

According to the value of the indicator n, as well as the case <i> (that is, λ = 0, which means n = 0), we can create a linear superposition. The analytical expression of $m_0$ is given as follows:

(6)

The authors would like to replace Equation (38):

$$c_n=2\cos \left(n\pi {\zeta }_0\right)$$

(38)

with

$$c_n=2\cos \left(n\pi {\zeta }_0\right),n=1,2,3,\cdots$$

(38)

(7)

Replacing the sentence in “Section 3.2. First-Order Concentration Moment Solution”:

If we can find the solution $w\left(\zeta ,t^{\ast };\tau \right)$ of Equation (51), then the first-order concentration moment $m_1\left(\zeta ,t^{\ast }\right)={\displaystyle {\int }_0^{t}w\left(\zeta ,t^{\ast };\tau \right)}d\tau$ can be obtained.

with

If we can find the solution $w\left(\zeta ,t^{\ast };\tau \right)$ of Equation (51), then the first-order concentration moment $m_1\left(\zeta ,t^{\ast }\right)={\displaystyle {\int }_0^{t^{\ast }}w\left(\zeta ,t^{\ast };\tau \right)}d\tau$ can be obtained.

(8)

Replacing the sentence in “Section 3.2. First-Order Concentration Moment Solution”:

Due to the boundary conditions of ${\left.w_{\zeta }\right|}_{\zeta =0}={\left.w_{\zeta }\right|}_{\zeta =1}=0$, $g_1\left(\tau \right)=0$ or $g_3\left(\tau \right)=0$. If $g_1(\tau )=0$, then $w\left(\zeta ,t^{\ast };\tau \right)=g_2\left(\tau \right)g_3\left(\tau \right)$, which contradicts the initial value condition; if $g_3\left(\tau \right)=0$, then $w\left(\xi ,t^{\ast };\tau \right)=0$, which does not meet the requirements, so <i> can be ruled out.

with

Due to the boundary conditions of ${\left.w_{\zeta }\right|}_{\zeta =0}={\left.w_{\zeta }\right|}_{\zeta =1}=0$, $g_1\left(\tau \right)\cdot g_3\left(\tau \right)=0$.

If $g_3(\tau )=0$, then $w\left(\zeta ,t^{\ast };\tau \right)=0$, which leads to a zero solution, so the case of $g_3\left(\tau \right)=0$ can be ruled out; so, $g_1(\tau )=0$, then, $w\left(\zeta ,t^{\ast };\tau \right)=c_0\left(\tau \right)$ (here, $c_0\left(\tau \right)=g_2\left(\tau \right)g_3\left(\tau \right),g_2\left(\tau \right)\ne 0,g_3\left(\tau \right)\ne 0$; otherwise, it leads to a zero solution). Thus, for $g(\tau )=0$, we obtain a function related only to $\tau$ (that is, $c_0\left(\tau \right)$, non-zero solution). It will be utilized in <iii> (for $g\left(\tau \right)<0$).

(9)

Replacing the sentence in “Section 3.2. First-Order Concentration Moment Solution”:

Because the coefficient determinant $\left|\begin{array}{cc}1& -1\\ e^{\sqrt{g\left(\tau \right)}}& -e^{-\sqrt{g\left(\tau \right)}}\end{array}\right|\ne 0$, we can resort to Kramer’s rule of linear algebra, and Equation (59) only has a zero solution, which means that $c_1\left(\tau \right)=c_2\left(\tau \right)=0$, ${\varphi }_1\left(\zeta ;\tau \right)=0$; this does not meet the requirements, so <ii> can be ruled out.

with

Because the coefficient determinant $\left|\begin{array}{cc}1& -1\\ e^{\sqrt{g\left(\tau \right)}}& -e^{-\sqrt{g\left(\tau \right)}}\end{array}\right|\ne 0$, we can resort to Kramer’s rule of linear algebra, and Equation (59) only has a zero solution, which means that $c_1\left(\tau \right)=c_2\left(\tau \right)=0$, ${\varphi }_1\left(\zeta ;\tau \right)=0$; this leads to a zero solution, so <ii> can be ruled out.

(10)

Authors would like to replace Equation (68):

Creating a linear superposition,

$$w\left(\zeta ,t^{\ast };\tau \right)={\displaystyle \sum _{n=1}^{\infty }{c}_n\left(\tau \right)e^{-n^2{\pi }^2{t}^{\ast }}\cos \left(n\pi \zeta \right)}$$

(68)

with

Considering the case <i> (that is, $g(\tau )=0$; it means n = 0), similar to Equation (34), we can create a linear superposition:

$$\begin{array}{ll}w\left(\zeta ,t^{\ast };\tau \right)& =c_0\left(\tau \right)+{\displaystyle \sum _{n=1}^{\infty }{c}_n\left(\tau \right)e^{-n^2{\pi }^2{t}^{\ast }}\cos \left(n\pi \zeta \right)}\\ & ={\displaystyle \sum _{n=0}^{\infty }{c}_n\left(\tau \right)e^{-n^2{\pi }^2{t}^{\ast }}\cos \left(n\pi \zeta \right)}\end{array}$$

(68)

(11)

The authors would like to replace Equation (69):

$$w\left(\zeta ;\tau \right)={\displaystyle \sum _{n=1}^{\infty }{c}_n\left(\tau \right)e^{-n^2{\pi }^2\tau }\cos \left(n\pi \zeta \right)}=\phi (\zeta ;\tau )$$

(69)

with

$$w\left(\zeta ;\tau \right)={\displaystyle \sum _{n=0}^{\infty }{c}_n\left(\tau \right)e^{-n^2{\pi }^2\tau }\cos \left(n\pi \zeta \right)}=\phi (\zeta ;\tau )$$

(69)

(12)

The authors would like to replace Equation (70):

$$\begin{array}{l}{c}_n\left(\tau \right)e^{-n^2{\pi }^2\tau }{\displaystyle {\int }_0^1{\cos }^2\left(n\pi \zeta \right)}d\zeta ={\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)}\phi \left(\zeta ;\tau \right)d\zeta ,\\ n=1,2,3,\cdots \end{array}$$

(70)

with

$$\begin{array}{l}{c}_0\left(\tau \right)={\displaystyle {\int }_0^1\phi \left(\zeta ;\tau \right)d\zeta ,}\\ c_n\left(\tau \right)e^{-n^2{\pi }^2\tau }{\displaystyle {\int }_0^1{\cos }^2\left(n\pi \zeta \right)}d\zeta ={\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)}\phi \left(\zeta ;\tau \right)d\zeta ,n=1,2,3,\cdots \end{array}$$

(70)

(13)

The authors would like to replace Equation (71):

Thus, we can obtain (n = 1, 2, 3,…)

$$\begin{array}{l}{c}_n\left(\tau \right)=2e^{n^2{\pi }^2\tau }{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)\phi \left(\zeta ;\tau \right)d\zeta },\\ w\left(\zeta ,t^{\ast };\tau \right)=2{\displaystyle \sum _{n=1}^{\infty }{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}\cos \left(n\pi \zeta \right)}{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)}\phi \left(\zeta ;\tau \right)d\zeta \end{array}$$

(71)

with

Thus, we can obtain

$$\left\{\begin{array}{l}{c}_0\left(\tau \right)={\displaystyle {\int }_0^1\phi \left(\zeta ;\tau \right)d\zeta }\\ c_n\left(\tau \right)=2e^{n^2{\pi }^2\tau }{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)\phi \left(\zeta ;\tau \right)d\zeta },n=1,2,3,\cdots \\ w\left(\zeta ,t^{\ast };\tau \right)={\displaystyle {\int }_0^1\phi \left(\zeta ;\tau \right)d\zeta }\\ +2{\displaystyle \sum _{n=1}^{\infty }{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}\cos \left(n\pi \zeta \right)}{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)}\phi \left(\zeta ;\tau \right)d\zeta \end{array}\right.$$

(71)

(14)

The authors would like to replace Equation (72):

$$\begin{array}{l}{m}_1\left(\zeta ,t^{\ast }\right)={\displaystyle {\int }_0^{t^{\ast }}w\left(\zeta ,t^{\ast };\tau \right)d\tau }\\ =2{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)\phi \left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\\ =2Pe{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)m_0\left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\\ =2Pe{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)}\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)\left(1+2{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi {\zeta }_0\right)\cos \left(n\pi \zeta \right)\cdot e^{-n^2{\pi }^2\tau }}\right)d\zeta }\right]d\tau \right\}\end{array}$$

(72)

with

$$\begin{array}{l}{m}_1\left(\zeta ,t^{\ast }\right)={\displaystyle {\int }_0^{t^{\ast }}w\left(\zeta ,t^{\ast };\tau \right)d\tau }\\ ={\displaystyle {\int }_0^{t^{\ast }}\left({\displaystyle {\int }_0^1\phi \left(\zeta ;\tau \right)d\zeta }\right)}d\tau +2{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)\phi \left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\\ =Pe{\displaystyle {\int }_0^{t^{\ast }}\left({\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right)m_0\left(\zeta ;\tau \right)d\zeta }\right)}d\tau +2Pe{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)m_0\left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\\ =Pe{\displaystyle {\int }_0^{t^{\ast }}\left[{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right)\left(1+2{\displaystyle \sum _{m=1}^{\infty }\cos \left(m\pi {\zeta }_0\right)\cos \left(m\pi \zeta \right)\cdot e^{-m^2{\pi }^2\tau }}\right)d\zeta }\right]}d\tau \\ +2Pe{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)}\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)\left(1+2{\displaystyle \sum _{m=1}^{\infty }\cos \left(m\pi {\zeta }_0\right)\cos \left(m\pi \zeta \right)\cdot e^{-m^2{\pi }^2\tau }}\right)d\zeta }\right]d\tau \right\}\end{array}$$

(72)

(15)

The authors would like to replace Equation (73):

$$\begin{array}{l}{m}_1\left(\zeta ,t^{\ast }\right)=2Pe{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right).\frac{\left(1-e^{-n^2{\pi }^2{t}^{\ast }}\right)}{n^2{\pi }^2}.\left\{{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)d\zeta }\right\}}\\ +4t^{\ast }Pe{\displaystyle \sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2{t}^{\ast }}\cos \left(n\pi {\zeta }_0\right)\cos \left(n\pi \zeta \right).\left\{{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right){\cos }^2\left(n\pi \zeta \right)d\zeta }\right\}}\\ +4Pe{\displaystyle \sum _{\begin{array}{l}m=1,\\ m\ne n\end{array}}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2{t}^{\ast }}\cos \left(m\pi {\zeta }_0\right)\cos \left(n\pi \zeta \right)}}{\displaystyle \frac{\left\{e^{{\pi }^2\left(n^2-m^2\right)t^{\ast }}-1\right\}}{\left(n^2-m^2\right){\pi }^2}}.\left\{{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right)\cos \left(n\pi \zeta \right)\cos \left(m\pi \zeta \right)d\zeta }\right\}\end{array}$$

(73)

with

$$\begin{array}{l}{m}_1\left(\zeta ,t^{\ast }\right)=2Pe{\displaystyle \sum _{m=1}^{\infty }\cos \left(m\pi {\zeta }_0\right).\frac{\left(1-e^{-m^2{\pi }^2{t}^{\ast }}\right)}{m^2{\pi }^2}.\left\{{\displaystyle {\int }_0^1\cos \left(m\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)d\zeta }\right\}}\\ +2Pe{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right).\frac{\left(1-e^{-n^2{\pi }^2{t}^{\ast }}\right)}{n^2{\pi }^2}.\left\{{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)d\zeta }\right\}}\\ +4t^{\ast }Pe{\displaystyle \sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2{t}^{\ast }}\cos \left(n\pi {\zeta }_0\right)\cos \left(n\pi \zeta \right).\left\{{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right){\cos }^2\left(n\pi \zeta \right)d\zeta }\right\}}\\ +4Pe{\displaystyle \sum _{\begin{array}{l}m=1,\\ m\ne n\end{array}}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2{t}^{\ast }}\cos \left(m\pi {\zeta }_0\right)\cos \left(n\pi \zeta \right)}}{\displaystyle \frac{\left\{e^{{\pi }^2\left(n^2-m^2\right)t^{\ast }}-1\right\}}{\left(n^2-m^2\right){\pi }^2}}.\left\{{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right)\cos \left(n\pi \zeta \right)\cos \left(m\pi \zeta \right)d\zeta }\right\}\end{array}$$

(73)

(16)

The authors would like to replace Equation (74):

$$\begin{array}{l}\Delta \eta =〈m_1(\zeta ,t^{\ast })〉={\displaystyle {\int }_0^1{m}_1(\zeta ,t^{\ast })d\zeta }\\ ={\left(-1\right)}^n{\displaystyle \frac{2Pe}{n\pi }}\left\{\begin{array}{l}{\displaystyle \sum _{n=1}^{\infty }\frac{\left(1-e^{-n^2{\pi }^2{t}^{\ast }}\right)}{n^2{\pi }^2}.\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)d\zeta }\right]}\\ +2t^{\ast }{\displaystyle \sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2{t}^{\ast }}\cos \left(n\pi {\zeta }_0\right).\left[{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right){\cos }^2\left(n\pi \zeta \right)d\zeta }\right]}\\ +2{\displaystyle \sum _{\begin{array}{l}m=1,\\ m\ne n\end{array}}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{e}^{-n^2{\pi }^2{t}^{\ast }}\cos \left(m\pi {\zeta }_0\right)}}{\displaystyle \frac{\left[e^{{\pi }^2\left(n^2-m^2\right)t^{\ast }}-1\right]}{\left(n^2-m^2\right){\pi }^2}}.\left[{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right)\cos \left(n\pi \zeta \right)\cos \left(m\pi \zeta \right)d\zeta }\right]\end{array}\right\}\end{array}$$

(74)

with

$$\begin{array}{l}\Delta \eta =〈m_1(\zeta ,t^{\ast })〉={\displaystyle {\int }_0^1{m}_1(\zeta ,t^{\ast })d\zeta }\\ =2Pe{\displaystyle \sum _{m=1}^{\infty }\cos \left(m\pi {\zeta }_0\right).\frac{\left(1-e^{-m^2{\pi }^2{t}^{\ast }}\right)}{m^2{\pi }^2}.\left\{{\displaystyle {\int }_0^1\cos \left(m\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)d\zeta }\right\}}\end{array}$$

(74)

(17)

Authors would like to replace Equation (76):

$$\begin{array}{l}{m}_p\left(\zeta ,t^{\ast }\right)=2{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)\left(Pe\cdot p{\psi }^{\prime }{m}_{p-1}+p\left(p-1\right)m_{p-2}\right)d\zeta }\right]d\tau \right\}}\\ =2Pe\cdot p{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }{m}_{p-1}\left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\\ +2p\left(p-1\right){\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)m_{p-2}\left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\end{array}$$

(76)

with

$$\begin{array}{l}{m}_p\left(\zeta ,t^{\ast }\right)={\displaystyle {\int }_0^{t^{\ast }}\left\{{\displaystyle {\int }_0^1\left[Pe\cdot p{\psi }^{\prime }{m}_{p-1}+p\left(p-1\right)m_{p-2}\right]d\zeta }\right\}d\tau }\\ +2{\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\left(\tau -t^{\ast }\right)}}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)\left(Pe\cdot p{\psi }^{\prime }{m}_{p-1}+p\left(p-1\right)m_{p-2}\right)d\zeta }\right]d\tau \right\}}\\ =Pe\cdot p\cdot \left\{{\displaystyle {\int }_0^{t^{\ast }}\left[{\displaystyle {\int }_0^1{\psi }^{\prime }\left(\zeta \right)m_{p-1}\left(\zeta ;\tau \right)d\zeta }\right]d\tau }\right\}+p\cdot \left(p-1\right)\cdot \left\{{\displaystyle {\int }_0^{t^{\ast }}\left[{\displaystyle {\int }_0^1{m}_{p-2}\left(\zeta ;\tau \right)d\zeta }\right]d\tau }\right\}\\ +2Pe\cdot p\cdot {\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)e^{-n^2{\pi }^2{t}^{\ast }}\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\tau }}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right){\psi }^{\prime }\left(\zeta \right)m_{p-1}\left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\\ +2p\cdot \left(p-1\right)\cdot {\displaystyle \sum _{n=1}^{\infty }\cos \left(n\pi \zeta \right)e^{-n^2{\pi }^2{t}^{\ast }}\left\{{\displaystyle {\int }_0^{t^{\ast }}{e}^{n^2{\pi }^2\tau }}\left[{\displaystyle {\int }_0^1\cos \left(n\pi \zeta \right)m_{p-2}\left(\zeta ;\tau \right)d\zeta }\right]d\tau \right\}}\end{array}$$

(76)

The authors and the Editorial Office would like to apologize for any inconvenience caused to the readers and state that the scientific conclusions are unaffected. These corrections were approved by the Academic Editor. The original article has been updated.