Let’s consider the following nonlinear algebraic equation,

$$\begin{array}{}\mathrm{N}\left[w\left(x,y\right)\right]=0,\end{array}$$(2)

where N and *w* (*x*, *y*) are a nonlinear differential operator and an unknown function, respectively. Using *q* *ϵ* [0, 1] as an embedding parameter in topology, the following zero-order deformation equation is constructed,

$$\begin{array}{}\left(1-q\right)\mathrm{L}\left[\psi \left(x,y;q\right)-{w}_{0}\left(x,y\right)\right]=q\hslash \mathrm{N}\left[\psi \left(x,y;q\right)\right],\end{array}$$(3)

where *ψ* (*x*, *y*; *q*) and *w*_{0}(*x*, *y*) are an unknown function and an initial approximation of *w*(*x*, *y*), respectively. It is to be noted that by setting *q* = 0 and *q* = 1, it holds,

$$\begin{array}{}\psi \left(x,y;0\right)={w}_{0}\left(x,y\right),\psi \left(x,y;1\right)=w\left(x,y\right),\end{array}$$(4)

respectively. As *q* varies from 0 to 1, *ψ* (*x*, *y*; *q*) varies from the initial approximation *w*_{0}(*x*, *y*) to the solution *w*(*x*, *y*). By expanding *ψ* (*x*, *y*; *q*) in a Taylor’s series with respect to *q*, the following homotopy-series is constructed,

$$\begin{array}{}\psi \left(x,y;q\right)={w}_{0}\left(x,y\right)+\sum _{i=1}^{+\mathrm{\infty}}{w}_{i}\left(x,y\right){q}^{i},\end{array}$$(5)

where,

$$\begin{array}{}{w}_{i}\left(x,y\right)=\frac{1}{i!}\frac{{\mathrm{\partial}}^{i}\psi \left(x,y;q\right)}{\mathrm{\partial}{q}^{i}}\left|{}_{q=0}\right..\end{array}$$(6)

If Eq. (5) converges at *q* = 1, one has,

$$\begin{array}{}\psi \left(x,y;q\right)={w}_{0}\left(x,y\right)+\sum _{i=1}^{+\mathrm{\infty}}{w}_{i}\left(x,y\right).\end{array}$$(7)

Differentiating the zero-order deformation Eq. (3) *i*-times with respect to *q*, dividing by *i*! and setting *q* = 1, the *i*th-order deformation equation will be constructed,

$$\begin{array}{}\mathrm{L}\left[{w}_{i}\left(x,y\right)-{\chi}_{i}{w}_{i-1}\left(x,y\right)\right]={\mathrm{R}}_{i}\left({w}_{i-1}\left(x,y\right)\right),\end{array}$$(8)

where,

$$\begin{array}{}{\mathrm{R}}_{i}\left({w}_{i-1}\left(x,y\right)\right)=\frac{1}{\left(i-1\right)!}\frac{{\mathrm{\partial}}^{i-1}\mathrm{N}\left[\psi \left(x,y;q\right)\right]}{\mathrm{\partial}{q}^{i-1}}\left|{}_{q=0}\right.,\end{array}$$(9)

and,

$$\begin{array}{}{\chi}_{i}=\left\{\begin{array}{ll}0,& i\le 1,\\ 1,& i>1.\end{array}\right.\end{array}$$(10)

To apply the HAM on the present problem, consider Eq. (1) and its corresponding boundary conditions given,

$$\begin{array}{}\psi \left(x,0;q\right)=0,\frac{\mathrm{\partial}\psi \left(x,y;q\right)}{\mathrm{\partial}y}\left|{}_{y=0}\right.=0.\end{array}$$(11)

It should be noted that the HAM enables us to choose an auxiliary linear operator [17]. To this end, the auxiliary linear operator L will be defined,

$$\begin{array}{}\mathrm{L}\left[\psi \left(x,y;q\right)\right]=\frac{{\mathrm{\partial}}^{2}\psi \left(x,y;q\right)}{\mathrm{\partial}{y}^{2}},\end{array}$$(12)

which has the property of,

$$\begin{array}{}\mathrm{L}\left[{b}_{0}+{b}_{1}y\right]=0,\end{array}$$(13)

where *b*_{0} and *b*_{1} are integration constants to be determined by the corresponding boundary conditions. Furthermore, the nonlinear differential operator N can be chosen in terms of Eq. (1),

$$\begin{array}{}\mathrm{N}\left[\psi \left(x,y;q\right)\right]=\frac{{\mathrm{\partial}}^{2}\psi \left(x,y;q\right)}{\mathrm{\partial}{x}^{2}}-x\frac{{\mathrm{\partial}}^{2}\psi \left(x,y;q\right)}{\mathrm{\partial}{y}^{2}}.\end{array}$$(14)

By solving the *i*th-order deformation Eq. (8), the corresponding *i*th-order approximated solution can be obtained,

$$\begin{array}{}{w}_{i}\left(x,y\right)={\displaystyle {\chi}_{i}{w}_{i-1}\left(x,y\right)+\hslash {\int}_{0}^{y}{\int}_{0}^{y}{\mathrm{R}}_{i}\left({w}_{i-1}\left(x,y\right)\right)dydy,}\end{array}$$(15)

and,

$$\begin{array}{}{\mathrm{R}}_{i}\left({w}_{i-1}\left(x,y\right)\right)=\frac{{\mathrm{\partial}}^{2}{w}_{i-1}\left(x,y\right)}{\mathrm{\partial}{x}^{2}}-x\frac{{\mathrm{\partial}}^{2}{w}_{i-1}\left(x,y\right)}{\mathrm{\partial}{y}^{2}}.\end{array}$$(16)

Therefore, the *i*th-order approximated solution of *w*(*x*, *y*) will be obtained,

$$\begin{array}{}{w}_{k}\left(x,y\right)\approx \sum _{i=0}^{k}{w}_{i}\left(x,y\right).\end{array}$$(17)

In theory, at the *p*th-order of approximation, the square residual error can be defined [28],

$$\begin{array}{}{\mathit{\Delta}}_{p}={\displaystyle \underset{0}{\overset{\mathrm{\infty}}{\int}}{\left(\mathrm{N}\left[\sum _{i=0}^{p}{w}_{i}\left(\xi \right)\right]\right)}^{2}d\xi ,}\end{array}$$(18)

where *ξ* = *ξ* (*x*, *y*). It should be noted that by decreasing the values of *Δ*_{p}, the convergence for corresponding series solution would be faster [28].

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