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 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:
(21)
where and . Let b, k, n and s be nonnegative integers and , then we get
(22)
so that, obviously,
By using the same method as in [15], we now give a higher-order partial differential equation for the generating function as follows.
We set
and
We have
By using Leibnitz’s formula for the v th derivative, with respect to x, of the product of the above two functions, we obtain the following higher-order partial differential equation:
By using (1) in the above partial differential equation, we get the following higher order partial differential equation:
By substituting (21) into the above equation, after some calculation, we arrive at the following theorem.
Theorem 7.1 Let . Let b, s and v be nonnegative integers with . Then we have
where
Integrating equation (22) (by parts) with respect to x from 0 to 1 and using Theorem 3.1, we have
for all b and s.
(23)
We recall from the work of Doha et al. [4] that if is a differentiable function of degree m and defined on , then a linear combination of the Bernstein polynomials can be written. Therefore, we can easily have
where , otherwise and
By using the same method as in [4], we write
where , otherwise . 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 ,
(24)
by the following boundary conditions:
(cf. [4]). By using the same method as that of Doha et al. [4], we apply the unification of the Bernstein-type polynomials to the Bernstein-Galerkin approximation for solving (24); that is,
and
By applying the Bernstein-Galerkin approximation (24), we find 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 on is defined by
By using this inner product, we modify (24) as follows:
(25)
where , and
The matrix representation of the above equation is given by
where
By using (23), one can easily find A, (); that is,
and
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 such that the linear system resulting in the Bernstein-Galerkin approximation to (25) is possible. That is,
where for all . The 2m 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:
subject to the boundary conditions , with the exact solution
(cf. see for detail [4, 16]).