# Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Ordinary and Partial Differential Equations

Authors: Ahmed Farooq Qasim; Ekhlass S. AL-Rawi
Journal: Journal of Applied Mathematics
 (2018)

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## 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.


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## Body

## 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 Section2, 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[0,1] are defined by (1)Bi,mx=mixi1-xm-iwhere the binomial coefficient is(2)mi=m!i!m-i!For example, when m=5, then the Bernstein terms are(3)B0,5x=1-x5B1,5x=5x1-x4B2,5x=10x21-x3B3,5x=10x31-x2B4,5x=5x41-xB5,5x=x5Definition 2 (Bernstein polynomials).
A linear combination of Bernstein basis polynomials(4)Bmx=∑i=0mBi,mxβiis called the Bernstein polynomials of degree m, where βi are the Bernstein coefficients.Definition 3.
Letf be a real valued function defined and bounded on [0,1]; let Bmf be the polynomial on [0,1], defined by(5)Bmf=∑i=0mmixi1-xm-ifimwhere Bmf is the m-th Bernstein polynomials for f(x).
For each functionf:0,1→R, we have(6)limm→∞⁡Bmfx=fxExample. If fx=ex,x∈0,1 then the Bernstein expanded for the function f(x) when m=5 is(7)Bmf=f01-x5+f155x1-x4+f2510x21-x3+f3510x31-x2+f455x41-x+f1x5Bmf=e01-x5+5e1/5x1-x4+10e2/5x21-x3+10e3/5x31-x2+5e4/5x41-x+e1x5In (1986) [14] Lorentz, prove that if the 2k-th order derivative f2k(x) is bounded in the interval (0,1) then for each x∈0,1(8)Bmfx=fx+∑a=22k-1faxa!maTm,ax+O1mkwhere(9)Tm,ax=∑kk-mxamkxk1-xm-kRemark (see [15]). Notice that Tm,ax is thea-th central moment of a random variable with a binomial appropriation with parameters m and x. Clearly, Tm,0=1, Tm,1=0. It is well known that the sequence Tm,ax satisfies the following recurrence:(10)Tm,a+1x=x1-xT′m,ax+maTm,a-1xIf we apply (8) tok = 1; 2; 3, then we obtain(11)Bmfx=fx+O1mBmfx=fx+x1-xf′′x2m+O1m2Bmfx=fx+x1-xf′′x2m+x1-x41-2xf3x+3x1-xf4x24m2+O1m3and higher level approximations can be computed.


## 3. ADM Based on Modified Bernstein Polynomials

Let us consider the following equation:(12)Lu+Nu+Ru=gxwhere L is an invertible linear term, N represents the nonlinear term, and R is the remaining linear part; from (12) we have(13)Lu=gx-Nu-Ru.Now, applying the inverse factorL-1 to both sides of (13) then via the initial conditions we find(14)u=fx-L-1Nu-L-1Ru,where L-1=∫0x(.)ds and f(x) are the terms having from integrating the rest of the termg(x) and from utilizing the given initial or boundary conditions. The ADM assumes thatN(u) (nonlinear term) can be decomposed by an infinite series of polynomials which is expressed in form (15)Nu=∑n=0∞Anuo,u1,…,unwhere An are the Adomian’s polynomials [16] defined as(16)An=1n!dndλnN∑i=0∞λiuiλ=0,n=0,1,2,…We expand the function g(x) by Bernstein series (17)gx=∑i=0maiBixwhere Bi(x) is the Bernstein polynomials.Now, using (14) and (17) we have(18)u0=L-1a0B0x+a1B1x+a2B2x+⋯.amBmx+θx,u1=-L-1Ru0-L-1Nu0,u2=-L-1Ru1-L-1Nu1,⋮and so on. These formulas are easy to compute by using Maple 13 software.In this paper, we improve the functiong(x) using modified Bernstein series(19)gx=∑i=0mmixi1-xm-ifim-∑a=22k-1faxa!maTm,axAnd we can approach the derivatives using the Bernstein polynomials(20)ddxBi,mx=mBi-1,m-1x-Bi,m-1x,Then (19) becomes(21)gx=∑i=0mmixi1-xm-ifim-∑a=22k-1da/dxaBi,mxa!maTm,axNow, using (18) and (21) we have(22)u0=L-1Bi,mx+θx,u1=-L-1Ru0-L-1Nu0,u2=-L-1Ru1-L-1Nu1,⋮The above equation is governing equation of ADM using modified Bernstein polynomials. The obtained approximate solution, ωVx=∑i=0Vui, 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 equation(23)d2ydt2+tdydt+t2y3=2+6t2et2+t2e3t2,y0=1,dydt0=0,with the exact solution (t)=et2. Using (12) we have(24)Ly+Ny+Ry=gxwhere L=d2/dt2,Ry=td/dt,Ny=t2y3, and gt=2+6t2et2+t2e3t2.
The Adomian polynomials for representing the nonlinear term Ny are(25)A0=t2y03,A1=t23y02y1,A2=t23y02y2+3y0y12,⋮Now L-1=∫0t∫0t.dtdt; then using (5) the classical Bernstein polynomials of g(t) when v=m=6 are(26)gbt=2+1.547324t+9.290164t2+7.83289t3+9.751887t4+7.659668t5+3.749864t6And modified Bernstein polynomials (21) of g(t) with k=2 are(27)gmbt=2-0.001037t+6.922082t2+1.997441t3+6.737662t4+11.051121t5+13.124523t6By (22), we have(28)y0=L-1gmbt+y0+dydt0t=1+t2-0.000173t3+0.57684t4+0.099872t5+0.224589t6+0.263122t7+0.234367t8,y1=-L-1tddty0-L-1A0=-0.25t4+0.000026t5-0.176912t6-0.011877t7+⋯,y2=-L-1tddty1-L-1A1=0.033333t6-0.000003t7+0.032348t8+0.011536t9+⋯,⋮And we obtain(29)ymbt=∑i=06yi=1+t2-0.000173t3+0.32684t4+⋯.The absolute error of ymbt and ybt is presented in Table 1 and Figure 1.Table 1
Comparison of absolute errors usingy6 when m=v=6 and k=2.
t y e x a c t - y b y e x a c t - y m b 0 0 0  0.01 2.580000 E-7 2.000000 E-9  0.02 2.068000 E-6 2.900000 E-8  0.03 6.992000 E-6 1.420000 E-7  0.04 1.660300 E-5 4.450000 E-7  0.05 3.249600 E-5 1.074000 E-6  0.06 5.629000 E-5 2.207000 E-6  0.07 8.962800 E-5 4.057000 E-6  0.08 1.341850 E-4 6.872000 E-6  0.09 1.916690 E-4 1.093000 E-5  0.1 2.638350 E-4 1.654900 E-5  MSE 1.369383957 E-8 4.632475500 E-11Figure 1
The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=6 and k=2.
(a) (b)Figure1 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 10-4 while the errors yielded from ADM with modified Bernstein polynomial are of 10-5.Example 2.
Consider the ordinary equation(30)d2ydt2+ydydt=tsin⁡2t2-4t2sin⁡t2+2cos⁡t2,0≤t≤1,y0=0,dydt0=0,with the exact solution (t)=sint2.
Here,L=d2/dt2, Ny=ydy/dt, and gt=tsin⁡2t2-4t2sin⁡t2+2cos⁡t2.
The Adomian polynomials for represent the nonlinear term Nu are(31)A0=y0ddty0,A1=y1ddty0+y0ddty1,A2=y2ddty0+y1ddty1+y0ddty1,⋮Then using (5) the classical Bernstein polynomials of g(t) when v=4 and m=16 is(32)gbt=20.00659171t+0.2233190t2+0.098085t3-3.39540t4+⋯And modified Bernstein polynomials (21) of g(t) with k=2 are(33)gmbt=2-0.007366t+0.2188855t2+1.389751t3-4.50213t4+⋯By (22), we have(34)ymbt=∑i=04yi=t2-0.001228t3+0.018241t4-0.030513t5+⋯,The absolute error of ymbt and ybt is presented in Table 2 and Figure 2.Table 2
Comparison of absolute errors usingy10 when m=16, v=4, and k=2.
t y e x a c t - y b y e x a c t - y m b 0 0 0  0.1 2.061126 E-6 3.074560 E-7  0.2 1.145239 E-5 1.058636 E-5  0.3 1.315732 E-5 5.114716 E-5  0.4 2.165603 E-4 1.331415 E-4  0.5 8.646500 E-4 2.420463 E-4  0.6 2.311555 E-3 3.299021 E-4  0.7 4.905204 E-3 3.231831 E-4  0.8 8.829753 E-3 1.540876 E-4  0.9 1.391114 E-2 1.870564 E-4  1 1.946226 E-2 6.088701 E-4  MSE 6.804632 E-5 7.217798344 E-8Figure 2
The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=v=10 and k=3.
(a) (b)Figure2 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 10-2 while the errors yielded from ADM with modified Bernstein polynomial are of 10-4.Example 3.
Consider the ordinary equation(35)dydt-ty+y2=et2,y0=1,with the exact solution (t)=et2/2.
HereL=d/dt, Ru=-ty,Ny=y2, and gt=et2.
Then using (5) the classical Bernstein polynomials of g(t) when v=8 and m=12 is(36)gbt=1+0.083623t+0.939177t2+0.197729t3+0.331572t4+⋯,and modified Bernstein polynomials (21) of g(t) with k=3 are(37)gmbt=1+0.003794t+0.957094t2+0.103509t3+0.410289t4+⋯.By (22), we have(38)umbt=∑i=08ui=1+0.501897t2-0.015567t3+0.159135t4+⋯,The absolute error of umbt and ubt is presented in Table 3 and Figure 3.
Figure3 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 10-4 while the errors yielded from ADM with modified Bernstein polynomial are of 10-6.Table 3
Comparison of absolute errors usingy10 when m=12, v=8, and k=3.
t y e x a c t - y b y e x a c t - y m b 0 0 0  0.01 4.134000 E-6 1.750000 E-7  0.02 1.635300 E-5 6.400000 E-7  0.03 3.639700 E-5 1.314000 E-6  0.04 6.402400 E-5 2.123000 E-6  0.05 9.900900 E-5 2.999000 E-6  0.06 1.411460 E-4 3.883000 E-6  0.07 1.902410 E-4 4.720000 E-6  0.08 2.461180 E-4 5.463000 E-6  0.09 3.086100 E-4 6.069000 E-6  0.1 3.775690 E-4 6.501000 E-6  MSE 3.699974901 E-8 1.619641710 E-11Figure 3
The absolute error between ADM with modified Bernstein polynomials and the exact solution when m=12, v=8, and k=3.
(a) (b)Example 4.
Consider the partial differential equation(39)∂y∂t+y∂y∂x-v∂2y∂x2=x2tcos⁡t2+sin2⁡t2,yx,0=0,Using (12) we have(40)Ly+Ny+Ry=gx,twhere L=d/dt, Ry=-v∂2y/∂x2, Ny=y∂y/∂x, and gx,t=x2tcos⁡t2+sin2t2.
The Adomian polynomials for Ny are(41)A0=yo∂yo∂x,A1=yo∂y1∂x+y1∂yo∂x,A2=yo∂y2∂x+y2∂yo∂x+y1∂y1∂x,⋮Then using (5) the classical Bernstein polynomials of g(x,t) when v=m=6 is(42)gbx,t=2.003859xt+0.103475xt2+0.149954xt3-0.27935xt4+⋯,and modified Bernstein polynomials (21) of g(x,t) with k=2 are(43)gmbx,t=1.986611xt+0.045744xt2+0.504281xt3-0.25375xt4+⋯.By (22) with v=1, we have(44)y0=L-1gmbx,t+yx,0=0.993306xt2+0.015248xt3+0.12607xt4-0.050750xt5…,y1=-L-1-v∂2y0∂x2-L-1A0=-0.197331xt5-0.005049xt6-0.035812xt7+0.012122xt8+⋯,y2=-L-1-v∂2y1∂x2-L-1A1=0.049003xt8+0.001783xt9+0.012105xt10-0.003795xt11+⋯,⋮And we obtain(45)ymbx,t=∑i=06yi=0.993306xt2+0.015248xt3+0.126070xt4-0.248081xt5-0.083749xt6+⋯,The absolute error of ymbx,t and ybx,t is presented in Table 4 and Figure 4 with the exact solution yx,t=xsint2.Table 4
Comparison of absolute errors usingy6 when m=v=6, k=2, and x=0.1.
t y e x a c t - y b y e x a c t - y m b 0 0 0  0.1 5.512102 E-6 4.149093 E-6  0.2 3.401160 E-5 1.849504 E-6  0.3 8.851342 E-5 2.838128 E-5  0.4 1.404426 E-5 9.121566 E-5  0.5 1.256008 E-4 1.620426 E-4  0.6 4.250973 E-5 1.880417 E-4  0.7 4.360788 E-4 1.086168 E-4  0.8 1.050197 E-3 1.020235 E-4  0.9 1.718854 E-3 3.752218 E-4  1 2.024768 E-3 4.863297 E-4  MSE 8.393551358 E-7 4.702782622 E-8Figure 4
The absolute error between ADM with classical and modified Bernstein polynomials and the exact solution usingy6 when m=v=6, k=2, and x=0.1.
(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 polynomialsAlso Figure4 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 10-3 while the errors yielded from ADM with modified Bernstein polynomial are of 10-4.


## 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 Tables1–4 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.



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