Our strategy is utilizing BCM to approximate the solutions *y*(*x*) where *y*(*x*) is given below. Define the Bezier polynomials of degree *n* over the interval [*x*_{0}, *x*_{f}] as follows:

$$\begin{array}{}{\displaystyle y(x)=\sum _{r=0}^{n}{a}_{r}{B}_{r,n}(\frac{x-{x}_{0}}{h}),\text{\hspace{0.17em}}{x}_{f}=1,\text{\hspace{0.17em}}{x}_{0}=0,}\end{array}$$(2)

where *h* = *x*_{f} – *x*_{0}, and

$$\begin{array}{}{\displaystyle {B}_{r,n}(\frac{x-{x}_{0}}{h})=\left(\genfrac{}{}{0ex}{}{n}{r}\right)\frac{1}{{h}^{n}}({x}_{f}-x{)}^{n-r}(x-{x}_{0}{)}^{r},}\end{array}$$(3)

is the Bernstein polynomial of degree *n* over the interval [*x*_{0}, *x*_{f}], *a*_{r} represents the control points. Suppose that *M*_{B} is the coefficient matrix of *B*_{r,n}(*x*), *r* = 0, 1, …, *n*, where *M*_{B}(*i*, *j*) is the coefficient of the *B*_{i,n}(*x*) with respect to the monomial *x*^{j–1}, then by Eq. (3) one may have

$$\begin{array}{}{\displaystyle {M}_{B}(i,j)=(-1{)}^{i+j}\left(\genfrac{}{}{0ex}{}{n}{i}\right)\left(\genfrac{}{}{0ex}{}{n-i}{j-i}\right),}\\ {\displaystyle \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=0,1,\dots ,n,j=i,\dots ,n,}\end{array}$$

By substituting *y*(*x*) in Eq. (1), one may define *R*_{1}(*x*, *a*_{0}, *a*_{1}, …, *a*_{n}) for *x* ∈ [*x*_{0}, *x*_{f}] as follows:

$$\begin{array}{}{\displaystyle {R}_{1}(x,{a}_{0},{a}_{1},\dots ,{a}_{n})}& =& {y}^{\u2033}-f(x,y,{y}^{\prime}).\end{array}$$(4)

The convergence of this method is proven where *n* → ∞ (see [9]).

Now, we define the residual function over the interval [*x*_{0}, *x*_{f}] as follows

$$\begin{array}{}{\displaystyle R=\underset{{x}_{0}}{\overset{{x}_{f}}{\int}}{\left({R}_{1}(x,{a}_{0},{a}_{1},\dots ,{a}_{n})\right)}^{2}dx.}\end{array}$$(5)

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.