Correction: Yusuf et al. Magneto-Bioconvection Flow of Williamson Nanofluid over an Inclined Plate with Gyrotactic Microorganisms and Entropy Generation. Fluids 2021, 6, 109

This is an erratum to our published paper Reference [1]. The aim of this note is to correct the typographical errors in our previous analysis. It is found that these errors have no effect on the obtained outcomes in Reference [1].

Notice of Erratum

This erratum pertains to elements that were written in error in the original article Reference [1]. The authors acknowledge Professor Asterios Pantokratoras Reference [2] from the Democritus University of Thrace in Greece for suggestions that led to these corrections. We also thank the editor-in-chief for suggesting a non-similar solution method.

In the paper mentioned above, there are some typographical errors that we want to correct in this erratum. The authors confirm that these errors and the new method of solution do not have any impact on the conclusions of the article and apologize for the inconvenience. The correct and final version is as follows:

The correct form of Equations (2)–(5) in Reference [1] is

$$\begin{array}{l}u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}=\nu \left({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+\sqrt{2}\mathrm{\Gamma }{\displaystyle \frac{\partial u}{\partial y}}{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right)-\left({\displaystyle \frac{\sigma B_0^2}{{\rho }_f}}+{\displaystyle \frac{\nu }{k_p}}\right)u+\\ {\displaystyle \frac{1}{{\rho }_f}}\left(\begin{array}{l}\left(1-{\displaystyle \frac{C_{\infty }}{{\delta }_C}}\right)g{\rho }_f{\mathrm{\Lambda }}_1\left(T-T_{\infty }\right)-g\left({\rho }_p-{\rho }_f\right)\left({\displaystyle \frac{C-C_{\infty }}{{\delta }_C}}\right)\\ -\left({\rho }_m-{\rho }_f\right){\mathrm{\Lambda }}_{2}g\left({\displaystyle \frac{N-N_{\infty }}{{\delta }_N}}\right)\end{array}\right)\cos \mathrm{\Omega },\end{array}$$

(2)

$$\begin{array}{l}{\left(\rho C_p\right)}_f\left(u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}\right)=k_f{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}-{\displaystyle \frac{\partial q_r}{\partial y}}+\left(\sigma B_0^2{u}^2+{\displaystyle \frac{\mu }{k_p}}{u}^2\right)\\ +\mu {\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2\left(1+{\displaystyle \frac{\mathrm{\Gamma }}{\sqrt{2}}}{\displaystyle \frac{\partial u}{\partial y}}\right)+{\left(\rho C_p\right)}_p\left({\displaystyle \frac{D_m}{{\delta }_C}}{\displaystyle \frac{\partial C}{\partial y}}{\displaystyle \frac{\partial T}{\partial y}}+{\displaystyle \frac{D_T}{T_{\infty }}}{\left({\displaystyle \frac{\partial T}{\partial y}}\right)}^2\right),\end{array}$$

(3)

$$u{\displaystyle \frac{\partial C}{\partial x}}+v{\displaystyle \frac{\partial C}{\partial y}}=D_m{\displaystyle \frac{{\partial }^{2}C}{\partial y^2}}+{\displaystyle \frac{{\delta }_C{D}_T}{T_{\infty }}}{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}-\alpha \left(C-C_{\infty }\right),$$

(4)

$$u{\displaystyle \frac{\partial N}{\partial x}}+v{\displaystyle \frac{\partial N}{\partial y}}+{\displaystyle \frac{bWc}{\left(C_f-C_{\infty }\right)}}\left({\displaystyle \frac{\partial }{\partial y}}\left(N{\displaystyle \frac{\partial C}{\partial y}}\right)\right)=D_n{\displaystyle \frac{{\partial }^{2}N}{\partial y^2}}.$$

(5)

Subject to the boundary conditions

$$\begin{array}{l}u=u_w=ax,v=0,-k_f{\displaystyle \frac{\partial T}{\partial y}}=d_t\left(T_f-T\right),\\ -D_m{\displaystyle \frac{\partial C}{\partial y}}=d_c\left(C_f-C\right),D_n{\displaystyle \frac{\partial N}{\partial y}}=d_n\left(N_f-N\right)\end{array}$$

(6)

$$u\to 0,T\to T_{\infty },C\to C_{\infty },N\to N_{\infty }.$$

(7)

The same corrections should be implemented in Equations (13)–(18) of Reference [1].

Now, introducing the following non-similar transformations,

$$u=ax{f}_{\eta },v=-\sqrt{{\nu }_{f}a}\left(f_{\xi }\xi +f\right),\xi ={\displaystyle \frac{x}{l}},\eta =\sqrt{{\displaystyle \frac{u_w}{{\nu }_{f}x}}}y,$$

$$\theta \left(\xi ,\eta \right)={\displaystyle \frac{\left(T-T_{\infty }\right)}{\left(T_f-T_{\infty }\right)}},\varphi \left(\xi ,\eta \right)={\displaystyle \frac{\left(C-C_{\infty }\right)}{\left(C_f-C_{\infty }\right)}},\psi \left(\xi ,\eta \right)={\displaystyle \frac{\left(N-N_{\infty }\right)}{\left(N_f-N_{\infty }\right)}},$$

(13)

which satisfies the continuity Equation (1), the boundary layer Equations (2)–(5) are reduced to the non-dimensional partial differential equations as below:

$$\begin{array}{l}{\displaystyle \frac{{\partial }^{3}f}{\partial {\eta }^3}}+\xi We{\displaystyle \frac{{\partial }^{2}f}{\partial {\eta }^2}}{\displaystyle \frac{{\partial }^{3}f}{\partial {\eta }^3}}-M{\displaystyle \frac{\partial f}{\partial \eta }}-k{\displaystyle \frac{\partial f}{\partial \eta }}-{\left({\displaystyle \frac{\partial f}{\partial \eta }}\right)}^2+f{\displaystyle \frac{{\partial }^{2}f}{\partial {\eta }^2}}+\\ {\displaystyle \frac{Gr}{{\mathrm{Re}}^2}}{\xi }^{-1}\left(\theta -A\varphi -Rb\psi \right)\cos \mathrm{\Omega }=\xi \left({\displaystyle \frac{\partial f}{\partial \eta }}{\displaystyle \frac{{\partial }^{2}f}{\partial \xi \partial \eta }}-{\displaystyle \frac{\partial f}{\partial \xi }}{\displaystyle \frac{{\partial }^{2}f}{\partial {\eta }^2}}\right),\end{array}$$

(14)

$$\begin{array}{l}(1+Rd){\displaystyle \frac{1}{\mathrm{Pr}}}{\displaystyle \frac{{\partial }^2\theta }{\partial {\eta }^2}}+Ec{\xi }^2\left(1+{\displaystyle \frac{We}{2}}{\displaystyle \frac{{\partial }^{2}f}{\partial {\eta }^2}}\right){\left({\displaystyle \frac{{\partial }^{2}f}{\partial {\eta }^2}}\right)}^2+Ec{\xi }^2\left(M+k\right){\left({\displaystyle \frac{\partial f}{\partial \eta }}\right)}^2+\\ f{\displaystyle \frac{\partial \theta }{\partial \eta }}+Nb{\displaystyle \frac{\partial \theta }{\partial \eta }}{\displaystyle \frac{\partial \varphi }{\partial \eta }}+Nt{\left({\displaystyle \frac{\partial \theta }{\partial \eta }}\right)}^2=\xi \left({\displaystyle \frac{\partial f}{\partial \eta }}{\displaystyle \frac{\partial \theta }{\partial \xi }}-{\displaystyle \frac{\partial f}{\partial \xi }}{\displaystyle \frac{\partial \theta }{\partial \eta }}\right),\end{array}$$

(15)

$${\displaystyle \frac{{\partial }^2\varphi }{\partial {\eta }^2}}+Sc\left(f{\displaystyle \frac{\partial \varphi }{\partial \eta }}-Kr\varphi \right)+{\displaystyle \frac{Nt}{Nb}}{\displaystyle \frac{{\partial }^2\theta }{\partial {\eta }^2}}=\xi Sc\left({\displaystyle \frac{\partial f}{\partial \eta }}{\displaystyle \frac{\partial \varphi }{\partial \xi }}-{\displaystyle \frac{\partial f}{\partial \xi }}{\displaystyle \frac{\partial \varphi }{\partial \eta }}\right),$$

(16)

$${\displaystyle \frac{{\partial }^2\psi }{\partial {\eta }^2}}-Pe\left({\displaystyle \frac{\partial \varphi }{\partial \eta }}{\displaystyle \frac{\partial \psi }{\partial \eta }}+({\delta }_1+\psi ){\displaystyle \frac{{\partial }^2\varphi }{\partial {\eta }^2}}\right)+Lef{\displaystyle \frac{\partial \psi }{\partial \eta }}=\xi Le\left({\displaystyle \frac{\partial f}{\partial \eta }}{\displaystyle \frac{\partial \psi }{\partial \xi }}-{\displaystyle \frac{\partial f}{\partial \xi }}{\displaystyle \frac{\partial \psi }{\partial \eta }}\right).$$

(17)

The transformed boundary conditions are

$$\begin{array}{l}{\displaystyle \frac{\partial f}{\partial \eta }}\left(\xi ,0\right)=1,f\left(\xi ,0\right)+\xi {\displaystyle \frac{\partial f}{\partial \xi }}\left(\xi ,0\right)=0,{\displaystyle \frac{\partial \theta }{\partial \eta }}\left(\xi ,0\right)=-B_1\left(1-\theta \right),\\ {\displaystyle \frac{\partial \varphi }{\partial \eta }}\left(\xi ,0\right)=-B_2\left(1-\varphi \right),{\displaystyle \frac{\partial \psi }{\partial \eta }}\left(\xi ,0\right)=-B_3\left(1-\psi \right),\end{array}$$

(18)

$${\displaystyle \frac{\partial f}{\partial \eta }}\left(\xi ,\infty \right)\to 0,\theta \left(\xi ,\infty \right)\to 0,\varphi \left(\xi ,\infty \right)\to 0,\psi \left(\xi ,\infty \right)\to 0.$$

(19)

The dimensionless parameters are as follows:

$$M={\displaystyle \frac{\sigma B_0^2}{\rho a}},k={\displaystyle \frac{\nu }{a{k}_p}},Gr=\left(1-{\displaystyle \frac{C_{\infty }}{{\delta }_C}}\right){\displaystyle \frac{g{\mathrm{\Lambda }}_1{l}^3\left(T_f-T_{\infty }\right)}{{\nu }^2}},Ec={\displaystyle \frac{a^2{l}^2}{Cp\left(T_f-T_{\infty }\right)}}$$

$$\mathrm{Re}={\displaystyle \frac{a{l}^2}{\nu }},A={\displaystyle \frac{Gr}{Gm}}={\displaystyle \frac{\left({\rho }_p-{\rho }_f\right)\left(C_f-C_{\infty }\right)}{\left({\delta }_C-C_{\infty }\right){\rho }_f{\mathrm{\Lambda }}_1\left(T_f-T_{\infty }\right)}},Rb={\displaystyle \frac{{\delta }_C\left({\rho }_m-{\rho }_f\right)}{\left({\delta }_C-C_{\infty }\right){\rho }_f}}{\displaystyle \frac{{\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_1\left(T_f-T_{\infty }\right)}}\left({\displaystyle \frac{N_f-N_{\infty }}{{\delta }_N}}\right)$$

$$Rd={\displaystyle \frac{16{\sigma }_{*}{T}_{\infty }^3}{3k_s{k}_f}},\mathrm{Pr}={\displaystyle \frac{{\left(\rho Cp\right)}_f\nu }{k_f}},\lambda ={\displaystyle \frac{Ge}{{\mathrm{Re}}^2}},B_1={\displaystyle \frac{d_t}{k_f}}\sqrt{{\displaystyle \frac{{\nu }_f}{a}}},B_2={\displaystyle \frac{d_c}{D_m}}\sqrt{{\displaystyle \frac{{\nu }_f}{a}}},B_3={\displaystyle \frac{d_n}{D_n}}\sqrt{{\displaystyle \frac{{\nu }_f}{a}}}$$

$$Nb={\displaystyle \frac{{\left(\rho Cp\right)}_p{D}_m\left(C_f-C_{\infty }\right)}{{\left(\rho Cp\right)}_f\nu {\delta }_c}},Nt={\displaystyle \frac{{\left(\rho Cp\right)}_p{D}_T\left(T_f-T_{\infty }\right)}{{\left(\rho Cp\right)}_f\nu T_{\infty }}},We=\sqrt{{\displaystyle \frac{2a^3}{\nu }}}l\mathrm{\Gamma }$$

(20)

Preliminaries of the Laguerre Collocation Method (LCM)

The highly nonlinear dimensionless partial differential equations (PDEs) given in Equations (14)–(17) of Reference [1] can be solved using the polynomial-based bivariate collocation method (see Reference [3]). In this section, the polynomial generating functions and general properties of the considered Laguerre polynomials are presented. Laguerre polynomials are obtained by the expression (see Reference [4])

$$L_m\left(\varsigma \right)={\displaystyle \frac{1}{m!}}{\left({\displaystyle \frac{d}{d\varsigma }}-1\right)}^m{\varsigma }^m$$

(1)

and the following are the recursive representations

$$\left(m+1\right)L_{m+1}\left(\varsigma \right)=\left(2m+1-\varsigma \right)L_m\left(\varsigma \right)-m{L}_{m-1}\left(\varsigma \right)$$

(2)

and

$$m{L}_m^{\prime }\left(\varsigma \right)=m{L}_m\left(\varsigma \right)-m{L}_{m-1}\left(\varsigma \right)$$

(3)

The first two terms, $L_0\left(\varsigma \right)$ and $L_1\left(\varsigma \right)$, are derived from Equation (1). The remaining terms are achieved using Equation (2) or Equation (3), and the corresponding few Laguerre polynomials are specified below:

$$\left.\begin{array}{l}{L}_0\left(\varsigma \right)=1,\\ L_1\left(\varsigma \right)=1-\varsigma ,\\ L_2\left(\varsigma \right)=1-2\varsigma +{\displaystyle \frac{1}{2}}{\varsigma }^2,\\ L_3\left(\varsigma \right)=1-3\varsigma +{\displaystyle \frac{3}{2}}{\varsigma }^2-{\displaystyle \frac{1}{6}}{\varsigma }^3,\\ L_4\left(\varsigma \right)=1-4\varsigma +3{\varsigma }^2-{\displaystyle \frac{5}{3}}{\varsigma }^3+{\displaystyle \frac{5}{24}}{\varsigma }^4,\\ L_5\left(\varsigma \right)=1-5\varsigma +5{\varsigma }^2-{\displaystyle \frac{2}{3}}{\varsigma }^3+{\displaystyle \frac{1}{22}}{\varsigma }^4-{\displaystyle \frac{1}{120}}{\varsigma }^5,\\ L_6\left(\varsigma \right)=1-6\varsigma +{\displaystyle \frac{135}{18}}{\varsigma }^2-{\displaystyle \frac{150}{45}}{\varsigma }^3+{\displaystyle \frac{90}{144}}{\varsigma }^4-{\displaystyle \frac{1}{20}}{\varsigma }^5+{\displaystyle \frac{1}{720}}{\varsigma }^6.\end{array}\right\}$$

(4)

To explain the application of the bivariate collocation method, the collocation method for a single variable function will be covered first, accompanied by the same procedure for a multivariate function. Now, the m^th-order boundary value problem (BVP)

$$G_m\left(\varsigma \right)y^{\left(m\right)}\left(\varsigma \right)+G_{\left(m-1\right)}\left(\varsigma \right)y^{\left(m-1\right)}\left(\varsigma \right)+\dots +G_1\left(\varsigma \right)y^1\left(\varsigma \right)+G_0\left(\varsigma \right)y^0\left(\varsigma \right)=h\left(\varsigma \right),$$

defined on the interval $[a^{*},b^{*}]$ with boundary conditions

$$y^{\left(m-1\right)}={\Delta }_1$$

(5)

is considered for illustrating the LCM procedure. The coefficients $G_m\left(\varsigma \right),\dots ,G_0\left(\varsigma \right)$ are constants or functions of $\varsigma$.

The following series $y\left(\varsigma \right)$ is taken to be the approximate solution of the BVP:

$$y\left(\varsigma \right)={\displaystyle \sum _{j=0}^N{\omega }_j}{L}_j\left(\varsigma \right)$$

(6)

Here, ${\omega }_j\left(j=0,1,\dots ,N\right)$ is the unknown coefficient. The substitution of Equation (6) in BVP yields the subsequent form:

$$\begin{array}{l}{G}_m\left(\varsigma \right){\displaystyle \sum _{j=0}^N{\omega }_j}{L}_j^{\left(m\right)}\left(\varsigma \right)+G_{\left(m-1\right)}\left(\varsigma \right){\displaystyle \sum _{j=0}^N{\omega }_j}{L}_j^{\left(m-1\right)}\left(\varsigma \right)+\dots +G_1\left(\varsigma \right){\displaystyle \sum _{j=0}^N{\omega }_j}{L}_j^1\left(\varsigma \right)+\\ G_0\left(\varsigma \right){\displaystyle \sum _{j=0}^N{\omega }_j}{L}_j\left(\varsigma \right)=h\left(\varsigma \right),\end{array}$$

(7)

Like coefficients ${\omega }_j$$\left(j=0,1,\dots ,N\right)$ are gathered for each expanded term in Equation (7) to produce

$${\displaystyle \sum _{j=0}^N{\omega }_j}\left(G_j^{*}\left(\varsigma \right)Q\left(\varsigma \right)\right)=h\left(\varsigma \right)$$

(8)

To solve ${\omega }_j$$\left(j=0,1,\dots ,N\right)$, $\kappa$ number of equations are produced using the boundary conditions, ${\displaystyle \frac{\kappa }{2}}$ each at the lower and higher bounds, respectively. Using Equation (8), the remaining $N-\kappa +1$ equations are produced at the collocation points. To produce the collocation points, the following equation is considered:

$${\varsigma }_i=a^{*}+{\displaystyle \frac{\left(b^{*}-a^{*}\right)i}{N-\left(\kappa -2\right)}},i=1,2,\dots ,N-\left(\kappa -1\right)$$

(9)

The different collocation points are obtained using Equation (9) and then substituted into Equation (8) to obtain the remaining $N-\kappa +1$ equations. Thus, Equation (8) can be rewritten as

$${\displaystyle \sum _{j=0}^N{\omega }_j}\left(G_j^{*}\left({\varsigma }_i\right)Q\left({\varsigma }_i\right)\right)=h\left({\varsigma }_i\right),\mathrm{for}{\omega }_j\left(j=0,1,\dots ,N\right)\mathrm{and}i=1,2,\dots ,N-\left(\kappa -1\right)$$

(10)

Solving all these equations provides the unknown coefficient values, and the substitution of these values in Equation (6) leads to the approximation solution of the considered BVP.

Solution by Bivariate Laguerre Collocation Method (BLCM)

The above procedure is extended to solve the equations with functions of two variables and is implemented for the dimensionless PDEs (Equations (14)–(17) of Reference [1]) using Laguerre polynomials by discretizing the equations at the collocation points of $\xi$ and $\eta$. The bivariate Laguerre collocation method (BLCM) assumes that the solution can be approximated by bivariate Laguerre polynomials, and thus, the following trail functions are approximated:

$$f\left(\eta ,\xi \right)={\displaystyle \sum _{i=0}^N{\displaystyle \sum _{j=0}^N{A}_{i,j}}{L}_i\left(\eta \right)L_j\left(\xi \right)}$$

$$\theta \left(\eta ,\xi \right)={\displaystyle \sum _{i=0}^N{\displaystyle \sum _{j=0}^N{B}_{i,j}}{L}_i\left(\eta \right)L_j\left(\xi \right)}$$

$$\varphi \left(\eta ,\xi \right)={\displaystyle \sum _{i=0}^N{\displaystyle \sum _{j=0}^N{F}_{i,j}}{L}_i\left(\eta \right)L_j\left(\xi \right)}$$

$$\psi \left(\eta ,\xi \right)={\displaystyle \sum _{i=0}^N{\displaystyle \sum _{j=0}^N{G}_{i,j}}{L}_i\left(\eta \right)L_j\left(\xi \right)}$$

(11)

where $L_i\left(\eta \right)$ and $L_j\left(\xi \right)$ are the Laguerre polynomials in $\xi$ and $\eta$ given by

$$\begin{array}{l}{L}_0\left(\eta \right)=1,\\ L_1\left(\eta \right)=1-\eta ,\\ L_2\left(\eta \right)=1-2\eta +{\displaystyle \frac{1}{2}}{\eta }^2,\\ L_3\left(\eta \right)=1-3\eta +{\displaystyle \frac{3}{2}}{\eta }^2-{\displaystyle \frac{1}{6}}{\eta }^3,\\ L_4\left(\eta \right)=1-4\eta +3{\eta }^2-{\displaystyle \frac{5}{3}}{\eta }^3+{\displaystyle \frac{5}{24}}{\eta }^4,\\ L_5\left(\eta \right)=1-5\eta +5{\eta }^2-{\displaystyle \frac{2}{3}}{\eta }^3+{\displaystyle \frac{1}{22}}{\eta }^4-{\displaystyle \frac{1}{120}}{\eta }^5,\\ L_6\left(\eta \right)=1-6\eta +{\displaystyle \frac{135}{18}}{\eta }^2-{\displaystyle \frac{150}{45}}{\eta }^3+{\displaystyle \frac{90}{144}}{\eta }^4-{\displaystyle \frac{1}{20}}{\eta }^5+{\displaystyle \frac{1}{720}}{\eta }^6.\end{array}$$

and

$$\begin{array}{l}{L}_0\left(\xi \right)=1,\\ L_1\left(\xi \right)=1-\xi ,\\ L_2\left(\xi \right)=1-2\xi +{\displaystyle \frac{1}{2}}{\xi }^2,\\ L_3\left(\xi \right)=1-3\xi +{\displaystyle \frac{3}{2}}{\xi }^2-{\displaystyle \frac{1}{6}}{\xi }^3,\\ L_4\left(\xi \right)=1-4\xi +3{\xi }^2-{\displaystyle \frac{5}{3}}{\xi }^3+{\displaystyle \frac{5}{24}}{\xi }^4,\\ L_5\left(\xi \right)=1-5\xi +5{\xi }^2-{\displaystyle \frac{2}{3}}{\xi }^3+{\displaystyle \frac{1}{22}}{\xi }^4-{\displaystyle \frac{1}{120}}{\xi }^5,\\ L_6\left(\xi \right)=1-6\xi +{\displaystyle \frac{135}{18}}{\xi }^2-{\displaystyle \frac{150}{45}}{\xi }^3+{\displaystyle \frac{90}{144}}{\xi }^4-{\displaystyle \frac{1}{20}}{\xi }^5+{\displaystyle \frac{1}{720}}{\xi }^6.\end{array}$$

Using the given boundary conditions, the corresponding equations are determined. Substitution of the approximated terms in Equations (14)–(17) of Reference [1] and thereafter discretizing them using the collocation points,

$${\eta }_i=a^{*}+{\displaystyle \frac{\left(b^{*}-a^{*}\right)i}{N-\left(\kappa -2\right)}},i=1,2,\dots ,N-\left(\kappa -1\right)$$

and

$${\xi }_i=a^{*}+{\displaystyle \frac{\left(b^{*}-a^{*}\right)i}{N-\left(\kappa -2\right)}},i=1,2,\dots ,N-\left(\kappa -1\right)$$

(12)

yields the remaining equations. The equations obtained by boundary conditions and collocation points form the $\left(N+1\right)\times \left(N+1\right)$ system of algebraic equations and are solved to obtain the values of the coefficients $A_{i,j}$, $B_{i,j}$, $F_{i,j}$, $G_{i,j}$, and $R_{i,j}$. Replacement of these values in Equation (11) generates the approximated solution for the fluid profiles.

Table 1. Numerical results of $f\left(\xi ,\eta \right)$ for various $\eta$ values.

$\mathit{\eta }$	$\mathit{f}\left(\mathit{\xi },\mathit{\eta }\right)$
	Similar Solution	Non-Similar Solution
0.10	0.0918059667	0.0910219622
0.12	0.1083097080	0.1072266029
0.14	0.1242338657	0.1228192445
0.16	0.1395940861	0.1378207843
0.18	0.1544054085	0.1522510162
2.00	0.1686822893	0.1661286993

is designed to show the relationship between the presented similar and non-similar solutions. The tabulated results agree moderately with each other, confirming the application of the non-similar solution using BLCM to the shown problem.

The authors gratefully acknowledge Professor Asterios Pantokratoras from the Democritus University of Thrace in Greece for the comments that led to these corrections, which have no effect on the main body of the article.