To perform the continuous function \(u(\mathbf{x})\) of *p*-Laplacian equations (1)–(3) via the SMRPI method, we have to outline it in this section. The continuous function \(u(\mathbf{x})\) can be represented via RBF \(\mathbf{R}_{i}(\mathbf{x})\) and monomials in the point of interest \(\mathbf{x} \in \Omega _{l}\). So, the coefficients \(a_{i}\) and \(b_{j}\) can be considered as

$$ u(\mathbf{x})=\sum_{i=1}^{n}R_{i}( \mathbf{x})a_{i}+\sum_{j=1}^{m}P_{j}( \mathbf{x})b_{j}=\mathbf{R}^{T}(\mathbf{x})\mathbf{a}+ \mathbf{P}^{T}( \mathbf{x})\mathbf{b}, $$

(4)

in which \(\Omega _{l}\) is a disk centered at \(x_{l}\) with radius \(r_{s}\), *n* is the number of point in \(\Omega _{l}\) and *m* is the number of polynomial basis functions. When \(m = 0\), only RBFs are used, otherwise, the RBF is augmented with *m* polynomial basis functions. In the point of interest \(x_{l}\), we enforce Eq. (4) to be satisfied at those *n* nodes surrounding it. Then the linear algebraic system of equations (4) is represented as follows:

$$ \mathbf{U}_{s}=\mathbf{R}^{T}_{n} \mathbf{a}+\mathbf{P}^{T}_{m} \mathbf{b}, $$

(5)

in which \(\mathbf{U}_{s}\) is the vector of function values defined as

$$ \mathbf{U}_{s}=\{u_{1},u_{2},u_{3}, \ldots ,u_{n}\}^{T}, $$

(6)

\(\mathbf{R}_{n} \) denotes the RBFs moment matrix as follows:

$$ \mathbf{R}_{n}= \begin{pmatrix} \mathbf{R}_{1,1} & \mathbf{R}_{1,2} & \cdots &\mathbf{R}_{1,n} \\ \vdots &\vdots &\ddots &\vdots \\ \mathbf{R}_{n,1} & \mathbf{R}_{n,2} & \cdots &\mathbf{R}_{n,n} \end{pmatrix}, $$

(7)

and \(\mathbf{P}_{m} \) represents the polynomial moment matrix defined by

$$ \mathbf{P}_{m}= \begin{pmatrix} p_{1}(\mathbf{x}_{1}) & p_{1}(\mathbf{x}_{2}) &\cdots ,& p_{1}( \mathbf{x}_{N}) \\ \vdots &\vdots &\ddots &\vdots \\ p_{m}(\mathbf{x}_{1}) & p_{m}(\mathbf{x}_{2}) & \cdots ,& p_{m}( \mathbf{x}_{N})\end{pmatrix} . $$

(8)

Also, the vector of unknown coefficients for RBFs is

$$ \mathbf{a}=\{a_{1},a_{2},a_{3}, \ldots ,a_{n}\}^{T}, $$

(9)

and the vector of unknown coefficients for the basis polynomial is

$$ \mathbf{b}=\{b_{1},b_{2},b_{3}, \ldots ,b_{m}\}^{T}. $$

(10)

Assume that \(r_{k} \), \(k=1,2,\ldots ,n \), being the distance between nodes in the support domain, \(\mathbf{R}_{k,i}=\mathbf{R}_{i}(r_{k})\) are the RBFs in (7). We added the following *m* equations in (5) to make a square matrix:

$$ \mathbf{P}^{T}_{m}\mathbf{a}=0. $$

(11)

So, the following system of equations is obtained from (5) and (11):

$$ \hat{\mathbf{U}}_{s}= \begin{pmatrix} \mathbf{U}_{s} \\ 0 \end{pmatrix} = \begin{pmatrix} \mathbf{R}_{n} & \mathbf{P}_{m} \\ \mathbf{P}^{T}_{m} & 0 \end{pmatrix} \begin{pmatrix} \mathbf{a} \\ \mathbf{b} \end{pmatrix} =\mathbf{G}\hat{\mathbf{a}_{s}}, $$

(12)

in which the matrix **G** is theoretically non-singular [29] and

$$ \hat{\mathbf{a}}_{s}= \begin{pmatrix} \mathbf{a} \\ \mathbf{b} \end{pmatrix} . $$

(13)

Now from (12) we have

$$ \hat{\mathbf{a}}_{s}=\mathbf{G}^{-1}\hat{ \mathbf{U}_{s}}. $$

(14)

By rewriting Eq. (4) we obtain

$$ u(\mathbf{x})=\hat{\Phi }(\mathbf{x})\hat{\mathbf{U}_{s}}, $$

(15)

such that

$$ \hat{\Phi }(\mathbf{x})=\bigl[\mathbf{R}_{n}( \mathbf{x}),\mathbf{P}_{m}( \mathbf{x})\bigr]\mathbf{G}^{-1}. $$

(16)

The shape functions corresponding to the nodal displacements of radial point interpolation method (RPIM), are the first *n* functions of the above vector and we show them by the vector \(\Phi ^{T}(\mathbf{x})\):

$$ \Phi (\mathbf{x})=\bigl\{ \phi _{1}(\mathbf{x}), \phi _{2}(\mathbf{x}), \ldots , \phi _{n}(\mathbf{x}) \bigr\} . $$

(17)

Now Eq. (15) converts to the following form:

$$ u(\mathbf{x})=\Phi (\mathbf{x})\mathbf{U}_{s}=\sum _{i=1}^{n}\phi _{i}( \mathbf{x})u_{i}. $$

(18)

Also it is well known that the Kronecker delta function property is attached to the RPIM shape functions by (16), which is explicitly written thus:

$$ \phi _{i}(\mathbf{x} _{j})= \textstyle\begin{cases} 1, & i=j, i,j=1,2, \ldots ,n, \\ 0, & i \neq j, i,j=1,2, \ldots ,n, \end{cases} $$

(19)

and lead to a sparse global collocation system. We assume that the total number of nodes that cover the \(\bar{\Omega }=\Omega \cap \partial \Omega \) is *N*. By rewriting Eq. (18), we have

$$ u(\mathbf{x})=\Phi (\mathbf{x})\mathbf{U}_{s}=\sum _{i=1}^{N}\phi _{i}( \mathbf{x})u_{i}. $$

(20)

Since corresponding to node \(x_{j}\) there is a shape function \(\phi _{i}(\mathbf{x})\), \(i=1,2, \ldots ,N\), obviously we have from Eq. (19)

$$ \forall x_{j} \in \Omega ^{c}_{x}, \quad \phi _{i}(x_{j})=0, $$

(21)

where \(\Omega ^{c}_{x}=\{x_{j}: x_{j} \notin \Omega _{x} \}\). Now the derivatives of \(u(\mathbf{x})\) respect to \(x_{i}\), *i*th component of \(\mathbf{x}=\{x_{1},\ldots ,x_{i},\ldots ,x_{N}\}\), determined as

$$ \frac{\partial u}{\partial x_{i}}(\mathbf{x})=\sum_{j=1}^{N} \frac{\partial \phi _{j}}{\partial x_{i}}(\mathbf{x})u_{j}, $$

(22)

and for high derivatives of \(u(\mathbf{x})\) we have

$$ \frac{\partial ^{s} u}{\partial (x_{i})^{s}}(\mathbf{x})=\sum_{j=1}^{N} \frac{\partial ^{s} \phi _{j}}{\partial (x_{i})^{s}}(\mathbf{x})u_{j}, $$

(23)

where \(\frac{\partial ^{s} }{\partial (x_{i})^{s}}\) is for the *s*th derivatives with respect to \(x_{i}\) implying that due to Eq. (21), \(\forall x_{j} \in \Omega _{x}^{c}\), \(\frac{\partial ^{s} \phi _{j}}{\partial (x_{i})^{s}}(\mathbf{x})=0\), \(s = 1, 2, \ldots \) . Denoting \(u^{(s)}_{x_{i}}(\cdot)=\frac{\partial ^{s} u(\cdot)}{\partial (x_{i})^{s}}\) and setting \(x = x_{i}\) in Eq. (20). Then the following matrix form is given:

$$ \begin{pmatrix} u^{(s)}_{x_{i}}(\mathbf{x}_{1}) \\ \vdots \\ u^{(s)}_{x_{i}}(\mathbf{x}_{N}) \end{pmatrix} = \overbrace{ \begin{pmatrix}\frac{\partial ^{s} \phi _{1}}{\partial (x_{i})^{s}}(\mathbf{x}_{1})&\cdots &\frac{\partial ^{s} \phi _{N}}{\partial (x_{i})^{s}}(\mathbf{x}_{1})\\ \vdots &\ddots &\vdots \\ \frac{\partial ^{s} \phi _{1}}{\partial (x_{i})^{s}}(\mathbf{x}_{N})&\cdots &\frac{\partial ^{s} \phi _{N}}{\partial (x_{i})^{s}}(\mathbf{x}_{N}) \end{pmatrix}}^{D_{x_{i}}^{s}} \begin{pmatrix} u_{1} \\ \vdots \\ u_{N} \end{pmatrix} . $$

(24)

This matrix-vector form for high-order derivatives is as follows:

$$ U^{s}_{x_{i}}=D_{x_{i}}^{s}U, $$

(25)

where

$$ U^{s}_{x_{i}}=\bigl\{ u^{(s)}_{x_{i}}( \mathbf{x}_{1}),\ldots ,u^{(s)}_{x_{i}}( \mathbf{x}_{N}) \bigr\} . $$

(26)