Abstract
In this paper, we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linear and nonlinear equations. This scheme is tested for four examples from ordinary and partial differential equations; furthermore, the obtained results demonstrate reliability and activity of the proposed technique. This strategy gives a precise and productive system in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and few emphasis prompts high exact solution. The numerical outcomes showed that the acquired estimated solutions were in appropriate concurrence with the correct solution.
1. Introduction
Adomian decomposition technique was established by George Adomian and has as of late turned into an extremely recognized strategy in connected sciences. The technique does not require any diminutiveness presumptions or linearization to solve the ordinary and partial differential equations and this produces the strategy extremely effective among alternate strategies. Recently, many iteration techniques have been used for solving nonlinear equations from ordinary, partial, and fractional equations [1], like variational iteration method and differential transform method [2], homotopy perturbation, and analysis methods [3]. Numerous works have been tested in various different regions, for example, warmth or mass exchange, incompressible fluid, nonlinear optics and gas elements wonders [4, 5], fractional Maxwell fluid [6, 7], and the Oldroyd-B fluid model [8].
The approximation used polynomials extremely important in scientific experiments where many rely on topics such as the study of statistics different population and the temperatures and others on the approximation theory. In addition, many experiments rely mainly on the approximate measurements and observations to be studied and processed by the appropriate scientific methods in order to reach the results expected from the study.
The Adomian decomposition technique is improved via Chebyshev polynomials in [9, 10], with Legendre polynomials [11] and with Laguerre polynomials [12].
This paper is organized as follows. In Section 2, the basic ideas of the modified Bernstein polynomials are described. Section 3 is devoted to solving a nonlinear differential equations using Adomian decomposition method based on modified Bernstein polynomials, the results and comparisons of the numerical solutions are presented in Section 4, and concluding remarks are given in Section 5.
2. The Modified Bernstein Polynomials
Polynomials are the mathematical technique as these can be characterized, figured, separated, and incorporated effortlessly. The Bernstein premise polynomials are trying to inexact the capacities. Bernstein polynomials are the better guess to a capacity with a couple of terms. These polynomials are utilized as a part of the fields of connected arithmetic and material science and PC helped geometric outlines and are likewise joined with different techniques like Galerkin and collocation technique to solve some differential and integral equations [13].
Definition 1 (Bernstein basis polynomials). The Bernstein basis polynomials of degree m over the interval are defined by where the binomial coefficient isFor example, when m=5, then the Bernstein terms are
Definition 2 (Bernstein polynomials). A linear combination of Bernstein basis polynomialsis called the Bernstein polynomials of degree m, where are the Bernstein coefficients.
Definition 3. Let be a real valued function defined and bounded on ; let be the polynomial on , defined bywhere is the m-th Bernstein polynomials for .
For each function , we have
Example. If then the Bernstein expanded for the function when m=5 isIn (1986) [14] Lorentz, prove that if the 2k-th order derivative is bounded in the interval (0,1) then for each whereRemark (see [15]). Notice that is the a-th central moment of a random variable with a binomial appropriation with parameters and . Clearly, , . It is well known that the sequence satisfies the following recurrence:If we apply (8) to k = 1; 2; 3, then we obtainand higher level approximations can be computed.
3. ADM Based on Modified Bernstein Polynomials
Let us consider the following equation:where is an invertible linear term, represents the nonlinear term, and is the remaining linear part; from (12) we haveNow, applying the inverse factor to both sides of (13) then via the initial conditions we findwhere ds and are the terms having from integrating the rest of the term g (x) and from utilizing the given initial or boundary conditions. The ADM assumes that N(u) (nonlinear term) can be decomposed by an infinite series of polynomials which is expressed in form where are the Adomian’s polynomials [16] defined asWe expand the function by Bernstein series where is the Bernstein polynomials.
Now, using (14) and (17) we haveand so on. These formulas are easy to compute by using Maple 13 software.
In this paper, we improve the function using modified Bernstein seriesAnd we can approach the derivatives using the Bernstein polynomialsThen (19) becomesNow, using (18) and (21) we haveThe above equation is governing equation of ADM using modified Bernstein polynomials. The obtained approximate solution, , by (22) has a comparison with the classic approximation solution and the correct solution.
4. Numerical Results
In this section, we solve ordinary and partial differential equations by ADM based on Bernstein polynomials and we compare with ADM based on classical Bernstein polynomial.
Example 1. Consider the ordinary equationwith the exact solution . Using (12) we havewhere , and .
The Adomian polynomials for representing the nonlinear term Ny areNow ; then using (5) the classical Bernstein polynomials of when v=m=6 areAnd modified Bernstein polynomials (21) of with k=2 areBy (22), we haveAnd we obtainThe absolute error of and is presented in Table 1 and Figure 1.
(a)
(b)
Figure 1 presents the absolute error of ADM with Bernstein polynomial in (a) and ADM with modified Bernstein polynomial in (b) at m=v=6 and k=2. The absolute errors generated using the ADM with Bernstein polynomial are of while the errors yielded from ADM with modified Bernstein polynomial are of .
Example 2. Consider the ordinary equationwith the exact solution .
Here, , , and .
The Adomian polynomials for represent the nonlinear term Nu areThen using (5) the classical Bernstein polynomials of when v=4 and m=16 isAnd modified Bernstein polynomials (21) of g(t) with k=2 areBy (22), we haveThe absolute error of and is presented in Table 2 and Figure 2.
(a)
(b)
Figure 2 presents the absolute error of ADM with Bernstein polynomial in (a) and ADM with modified Bernstein polynomial in (b) at m=v=10 and k=3. The absolute errors generated using the ADM with Bernstein polynomial are of while the errors yielded from ADM with modified Bernstein polynomial are of .
Example 3. Consider the ordinary equationwith the exact solution .
Here , , and .
Then using (5) the classical Bernstein polynomials of when v=8 and m=12 isand modified Bernstein polynomials (21) of with k=3 areBy (22), we haveThe absolute error of and is presented in Table 3 and Figure 3.
Figure 3 presents the absolute error of ADM with Bernstein polynomial in (a) and ADM with modified Bernstein polynomial in (b) at m=12, v=8, and k=3. The absolute errors generated using the ADM with Bernstein polynomial are of while the errors yielded from ADM with modified Bernstein polynomial are of .
(a)
(b)
Example 4. Consider the partial differential equationUsing (12) we havewhere , , , and .
The Adomian polynomials for Ny areThen using (5) the classical Bernstein polynomials of when v=m=6 isand modified Bernstein polynomials (21) of with k=2 areBy (22) with , we haveAnd we obtainThe absolute error of and is presented in Table 4 and Figure 4 with the exact solution .
(a) The absolute error for classical Bernstein polynomials
(b) The numerical solution for classical Bernstein polynomials
(c) The absolute error for modified Bernstein polynomials
(d) The numerical solution for modified Bernstein polynomials
Also Figure 4 presents the absolute error of ADM with Bernstein polynomial in (a) and ADM with modified Bernstein polynomial in (c) at m=v=6 and k=2. The absolute errors generated using the ADM with Bernstein polynomial are of while the errors yielded from ADM with modified Bernstein polynomial are of .
5. Conclusions
In this paper, we show that utilizing modified Bernstein polynomials is smartly thought to modify the performance of the Adomian decomposition technique. The fundamental preferred standpoint of this strategy is that it can be used specifically for all sort of differential and integral equations.
We utilize modified Bernstein extensions of the nonlinear term to get more exact outcomes. Figures empower us to consider the difference between utilizing two strategies graphically. Tables are additionally given to demonstrate the variety of the outright mistakes for bigger estimation, to be specific for bigger m. We observed from the numerical outcomes in Tables 1–4 and Figures 1–4 that the ADM with modification Bernstein polynomials gives more exact and robust numerical solution than the classical Bernstein polynomials. Every one of the calculations was done with the guide of Maple 13 programming.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research is supported by College of Computer Sciences and Mathematics, University of Mosul, Republic of Iraq, under Project no. 11251357.