In [4], Doha *et al*. gave an application of the Bernstein polynomials for solving high even-order differential equations by using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods. The methods do not contain generating functions for proving explicitly the derivatives formula of the Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves. Here, we prove this formula for the unification of the Bernstein-type polynomials ${\mathfrak{S}}_{n}(b,s,x)$ by a higher-order partial differential equation and functional equations. We also give some remarks and applications related to these polynomials and the Bernstein-Galerkin method.

We modify (1) as follows:

$\begin{array}{rcl}\mathcal{F}(t,b-k,s-k:x)& =& \frac{{2}^{b-k}{x}^{bs-k}{(\frac{t}{2})}^{bs-k}{e}^{t(1-x)}}{(bs-k)!}\\ =& \sum _{n=0}^{\mathrm{\infty}}{\mathfrak{S}}_{n}(b-k,s-k,x)\frac{{t}^{n}}{n!},\end{array}$

(21)

where

$k\in {\mathbb{N}}_{0}=\{0,1,2,\dots \}$ and

$x\in [0,1]$. Let

*b*,

*k*,

*n* and

*s* be nonnegative integers and

$n\ge bs-k\ge 0$, then we get

${\mathfrak{S}}_{n}(b-k,s-k,x)=\left(\genfrac{}{}{0ex}{}{n}{bs-k}\right)\frac{{x}^{bs-k}{(1-x)}^{n-bs+k}}{{2}^{b(s-1)}},$

(22)

so that, obviously,

${\mathfrak{S}}_{n}(b,s,x)={\mathfrak{S}}_{n}(b-0,s-0,x).$

By using the same method as in [15], we now give a higher-order partial differential equation for the generating function $\mathcal{F}(t,b,s:x)$ as follows.

We set

$g(t,x;b,s)=\frac{{2}^{b}{x}^{bs}{(\frac{t}{2})}^{bs}}{(bs)!}$

We have

$\mathcal{F}(t,b,s:x)=g(t,x;b,s)h(t,x).$

By using Leibnitz’s formula for the

*v* th derivative, with respect to

*x*, of the product

$\mathcal{F}(t,b,s:x)$ of the above two functions, we obtain the following higher-order partial differential equation:

$\frac{{\partial}^{v}\mathcal{F}(t,b,s:x)}{\partial {x}^{v}}=\sum _{j=0}^{v}\left(\genfrac{}{}{0ex}{}{v}{j}\right)\left(\frac{{\partial}^{j}g(t,x;b,s)}{\partial {x}^{j}}\right)\left(\frac{{\partial}^{v-j}h(t,x)}{\partial {x}^{v-j}}\right).$

By using (1) in the above partial differential equation, we get the following higher order partial differential equation:

$\frac{{\partial}^{v}\mathcal{F}(t,b,s:x)}{\partial {x}^{v}}=\sum _{j=0}^{v}{(-1)}^{v-j}\left(\genfrac{}{}{0ex}{}{v}{j}\right)\mathcal{F}(t,b-j,s-j:x).$

By substituting (21) into the above equation, after some calculation, we arrive at the following theorem.

**Theorem 7.1** *Let* $x\in [a,b]$.

*Let* *b*,

*s* *and* *v* *be nonnegative integers with* $n\ge bs$.

*Then we have*${\mathfrak{S}}_{n}^{(v)}(b,s,x)=\frac{n!}{(n-v)!}\sum _{j=0}^{v}{(-1)}^{v-j}\left(\genfrac{}{}{0ex}{}{v}{j}\right){\mathfrak{S}}_{n-v}(b-j,s-j,x),$

*where*
${\mathfrak{S}}_{n}^{(v)}(b,s,x)=\frac{{d}^{v}{\mathfrak{S}}_{n}(b,s,x)}{d{x}^{v}}.$

Integrating equation (

22) (by parts) with respect to

*x* from 0 to 1 and using Theorem 3.1, we have

${\int}_{0}^{1}{\mathfrak{S}}_{n}(b,s,x)\phantom{\rule{0.2em}{0ex}}dx=\frac{{2}^{b(1-s)}}{n+1},$

for all

*b* and

*s*.

${\int}_{0}^{1}{\mathfrak{S}}_{n}^{(v)}(b,s,x){\mathfrak{S}}_{n}(b,s,x)\phantom{\rule{0.2em}{0ex}}dx=\frac{{2}^{2b(1-s)}n!}{(2n+1)(n-v)!}\left(\genfrac{}{}{0ex}{}{n}{bs}\right)\sum _{j=0}^{v}{(-1)}^{v-j}\frac{\left(\genfrac{}{}{0ex}{}{v}{j}\right)\left(\genfrac{}{}{0ex}{}{n}{bs-j}\right)}{\left(\genfrac{}{}{0ex}{}{2n}{2bs-j}\right)}.$

(23)

We recall from the work of Doha

*et al*. [

4] that if

$f(x)$ is a differentiable function of degree

*m* and defined on

$[0,1]$, then a linear combination of the Bernstein polynomials can be written. Therefore, we can easily have

$f(x)=\sum _{b,s=0}^{m}{c}_{b,s,m}{\mathfrak{S}}_{m}(b,s,x),$

where

$bs\le m$, otherwise

${\mathfrak{S}}_{m}(b,s,x)=0$ and

$\sum _{b,s=0}^{m}=\sum _{b=0}^{m}\sum _{s=0}^{m}.$

By using the same method as in [

4], we write

${f}^{(v)}(x)=\frac{{d}^{v}f(x)}{d{x}^{v}}=\sum _{b,s=0}^{m}{c}_{b,s,m}\frac{{d}^{v}{\mathfrak{S}}_{m}(b,s,x)}{d{x}^{v}},$

where

$bs\le m$, otherwise

${\mathfrak{S}}_{m}(b,s,x)=0$. We now give an application for the solution of high even-order differential equations. We also recall from the work of Doha

*et al*. [

4] that for

$x\in [0,1]$,

$f(x)={u}^{(2m)}+\sum _{j=1}^{2m-1}{\gamma}_{j}{u}^{(j)}+{\gamma}_{0}u$

(24)

by the following boundary conditions:

${u}^{(v)}(0)=0,\phantom{\rule{2em}{0ex}}{u}^{(v)}(1)=0;\phantom{\rule{1em}{0ex}}0\le v\le m-1$

(

*cf*. [

4]). By using the same method as that of Doha

*et al*. [

4], we apply the unification of the Bernstein-type polynomials

${\mathfrak{S}}_{m}(b,s,x)$ to the Bernstein-Galerkin approximation for solving (24); that is,

${Y}_{m}=\{{\mathfrak{S}}_{m}(b,s,x):m\ge bs\}$

and

${Z}_{m}=\{u\in {Y}_{m}:{u}^{(v)}(0)=0,{u}^{(v)}(1)=0;0\le v\le m-1\}.$

By applying the Bernstein-Galerkin approximation (24), we find ${u}_{m}\in {Z}_{m}$ as follows. For solving this equation, we need the following notations, which we recall from the work of Doha *et al*. [[4], p.9, Eq. (4.4)].

The inner product

$\u3008u,v\u3009$ on

${L}^{2}(I)$ is defined by

$\u3008u,v\u3009={\int}_{I}u(x)v(x)\phantom{\rule{0.2em}{0ex}}dx.$

By using this inner product, we modify (24) as follows:

$\u3008f(x),{\mathfrak{g}}_{n}(k,x)\u3009=\u3008{u}_{n}^{(2m)},{\mathfrak{g}}_{n}(k,x)\u3009+\sum _{j=1}^{2m-1}{\gamma}_{j}\u3008{u}_{n}^{(j)},{\mathfrak{g}}_{n}(k,x)\u3009+{\gamma}_{0}\u3008{u}_{n},{\mathfrak{g}}_{n}(k,x)\u3009,$

(25)

where

$m\le k\le n-m$,

$2m\le n$ and

${\mathfrak{g}}_{n}(k,x)={\mathfrak{S}}_{n}(b+k,s+k,x)=\left(\genfrac{}{}{0ex}{}{n}{bs+k}\right)\frac{{x}^{bs+k}{(1-x)}^{n-bs-k}}{{2}^{b(s-1)}}.$

The matrix representation of the above equation is given by

$F=(A+\sum _{j=1}^{2m-1}{\gamma}_{j}{B}_{j}+{\gamma}_{0}{B}_{0})C,$

By using (23), one can easily find

*A*,

${B}_{j}$ (

$j=0,1,2,\dots ,2m-1$); that is,

${a}_{kj}=\u3008{\mathfrak{g}}_{n}^{(2m)}(j,x),{\mathfrak{g}}_{n}(k,x)\u3009={\int}_{0}^{1}{\mathfrak{g}}_{n}^{(2m)}(j,x){\mathfrak{g}}_{n}(k,x)\phantom{\rule{0.2em}{0ex}}dx$

and

${b}_{kj}^{i}=\u3008{\mathfrak{g}}_{n}^{(i)}(j,x),{\mathfrak{g}}_{n}(k,x)\u3009={\int}_{0}^{1}{\mathfrak{g}}_{n}^{(i)}(j,x){\mathfrak{g}}_{n}(k,x)\phantom{\rule{0.2em}{0ex}}dx.$

**Remark 7.2** According to Doha

*et al.* [

4], it is important to apply the Galerkin-spectral Bernstein approximation for how to choose an appropriate basis for

${Z}_{m}$ such that the linear system resulting in the Bernstein-Galerkin approximation to (25) is possible. That is,

${Z}_{m}=span\{{\mathfrak{g}}_{n}(m,x),{\mathfrak{g}}_{n}(m+1,x),\dots ,{\mathfrak{g}}_{n}(n-m,x)\},$

where ${\mathfrak{g}}_{n}(k,x)\in {Z}_{m}$ for all $m\le k\le n-m$. The 2*m* boundary conditions lead to the first *m*, and the least *m* expansion coefficients are zero.

**Remark 7.3** By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, Doha

*et al.* [

4] solved the following boundary value problem:

${u}^{(2)}(x)-u(x)=(4-2{x}^{2})sinx+4xcosx,\phantom{\rule{1em}{0ex}}x\in [0,1],$

subject to the boundary conditions

$u(0)=u(1)=0$, with the exact solution

(*cf*. see for detail [4, 16]).