The function *f* is approximated by Boubaker polynomials as following:

$$f(x)\simeq {\displaystyle \sum _{i=0}^{N}{c}_{i}}{B}_{i}={C}^{T}B(x),$$(10)

where **B**(*x*)^{T} = [*B*_{0}, *B*_{1}, …, *B*_{N}], *B*_{i}(*x*), *i* = 0, 1, 2, ···, *N* denote the Boubaker polynomials, *C*^{T} = [*c*_{0}, *c*_{1}, …, *c*_{N}] are unknown Boubaker cofficients and *N* is chosen as any positive integer.

Then *C*^{T} can be obtained by

$${C}^{T}\u3008B(x),B(x)\u3009=\u3008f,B(x)\u3009,$$(11)

where

$$\u3008f,B(x)\u3009={\displaystyle \underset{0}{\overset{1}{\int}}f}(x)B{(x)}^{T}dx=[\u3008f,{B}_{0}\u3009,\u3008f,{B}_{1}\u3009,\dots ,\u3008f,{B}_{m}\u3009],$$(12)

and 〈**B**(*x*), **B**(*x*)〉 is called the dual matrix of *ϕ* denoted by *Q*, and *Q* is obtained as:

$$Q=\u3008B(x),B(x)\u3009={\displaystyle \underset{0}{\overset{1}{\int}}B}(x)B{(x)}^{T}dx,$$(13)

and then

$${C}^{T}=\left({\displaystyle \underset{0}{\overset{1}{\int}}f}(x)B{(x)}^{T}dx\right){Q}^{-1}.$$(14)

By using the expression (7) and taking *n* = 0, …, *N*, we find the corresponding matrix relation as we can express Boubaker polynomials in terms of power basis functions

$$B(x)=ZX(x),$$(15)

where

$$X(x)={\left[1\hspace{0.17em}\hspace{0.17em}x\cdots {x}^{N}\right]}^{T},$$(16)

and if *N* is odd,

$$\begin{array}{l}Z=\hfill \\ \left[\begin{array}{ccccccc}{\phi}_{0,0}& 0& 0& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& 0\\ 0& {\phi}_{1,0}& 0& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& 0\\ {\phi}_{2,1}& 0& {\phi}_{2,0}& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \hspace{0.17em}\hspace{0.17em}& \vdots \\ {\phi}_{N-1,\frac{N-1}{2}}& 0& {\phi}_{N-1,\frac{N-3}{2}}& 0& \cdots \hspace{0.17em}\hspace{0.17em}& {\phi}_{N-1,0}& 0\\ 0& {\phi}_{N,\frac{N-1}{2}}& 0& {\phi}_{N,\frac{N-3}{2}}& \cdots \hspace{0.17em}\hspace{0.17em}& 0& {\phi}_{N,0}\end{array}\right]\hfill \end{array}$$

(default) and if *N* is even,

$$\begin{array}{l}Z=\hfill \\ \left[\begin{array}{ccccccc}{\phi}_{0,0}& 0& 0& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& 0\\ 0& {\phi}_{1,0}& 0& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& 0\\ {\phi}_{2,1}& 0& {\phi}_{2,0}& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \hspace{0.17em}\hspace{0.17em}& \vdots \\ 0& {\phi}_{N-1,\frac{N-1}{2}}& 0& {\phi}_{N-1,\frac{N-4}{2}}& \cdots \hspace{0.17em}\hspace{0.17em}& {\phi}_{N-1,0}& 0\\ {\phi}_{N,\frac{N}{2}}& 0& {\phi}_{N,\frac{N-2}{2}}& 0& \cdots \hspace{0.17em}\hspace{0.17em}& 0& {\phi}_{N,0}\end{array}\right]\hfill \end{array}$$

where

$$\begin{array}{l}{B}_{n}(x)={\displaystyle \sum _{p=0}^{\xi (n)}{\phi}_{n,p}}{x}^{n-2p},\hfill \\ n=0,1,\cdots ,N,\text{\hspace{1em}}p=0,1,\cdots ,\lfloor \frac{n}{2}\rfloor .\hfill \end{array}$$(17)

$${\phi}_{n,p}=\left[\frac{(n-4p)}{(n-p)}{C}_{n-p}^{p}\right]{(-1)}^{p}.$$(18)

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