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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 16, Issue 1

Issues

Volume 13 (2015)

Alternative methods for solving nonlinear two-point boundary value problems

Fateme Ghomanjani / Stanford Shateyi
Published Online: 2018-06-26 | DOI: https://doi.org/10.1515/phys-2018-0050

Abstract

In this sequel, the numerical solution of nonlinear two-point boundary value problems (NTBVPs) for ordinary differential equations (ODEs) is found by Bezier curve method (BCM) and orthonormal Bernstein polynomials (OBPs). OBPs will be constructed by Gram-Schmidt technique. Stated methods are more easier and applicable for linear and nonlinear problems. Some numerical examples are solved and they are stated the accurate findings.

Keywords: Nonlinear boundary value problems; orthonormal Bernstein

PACS: 02.60.Cb; 02.60.Lj

1 Introduction

In this paper, the following NTBVPs are considered

y=f(x,y,y),lay(a)may(a)=ka,x[x0,xf]=[a,b],a,bR,lby(b)+mby(b)=kb,(1)

where f ∈ (C[a, b] × R2, R) is a nonlinear function, la, ma, lb, and mb are given positive constants and ka, kb are given constants.

We consider the above NTBVPs which occur in applied mathematics, theoretical physics, engineering, control and optimization theory. By various researchers, different numerical methods have been proposed. Among these are the Adomian decomposition method (ADM) [1], He-Laplace method [2], Homotopy perturbation method (HPM) [3]. There are many methods for solving of two-point boundary value problems ODEs.

Additionally, many papers stated the BCM. In Ref. [5], authors utilized the BCM for solving delay differential equation (DDE) and optimal control of switched systems numerically. In Ref. [6], authors stated the utilization of BCM on some linear optimal control systems with pantograph delays. For solving the quadratic Riccati differential equation BCM is utilized (see Ref. [7]). In Ref. [8], authors presented BCM for solving singular nonlinear equations of Lane-Emden type. In this work, one may utilize BCM and orthonormal Bernstein polynomial (OBP) method for solving NTBVPs.

The Bernstein polynomials are not orthonormal therefore Gram-Schmidt technique could be utilized to construct OBP.

The organization of this paper is classified as follows: Problem statement is introduced in Section 2. Orthonormal Bernstein polynomials of arbitrary order are presented in Section 3. Numerical examples are solved in Section 4. Finally, Section 5 will give a conclusion briefly.

2 Problem statement

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 [x0, xf] as follows:

y(x)=r=0narBr,n(xx0h),xf=1,x0=0,(2)

where h = xfx0, and

Br,n(xx0h)=nr1hn(xfx)nr(xx0)r,(3)

is the Bernstein polynomial of degree n over the interval [x0, xf], ar represents the control points. Suppose that MB is the coefficient matrix of Br,n(x), r = 0, 1, …, n, where MB(i, j) is the coefficient of the Bi,n(x) with respect to the monomial xj–1, then by Eq. (3) one may have

MB(i,j)=(1)i+jniniji,i=0,1,,n,j=i,,n,

By substituting y(x) in Eq. (1), one may define R1(x, a0, a1, …, an) for x ∈ [x0, xf] as follows:

R1(x,a0,a1,,an)=yf(x,y,y).(4)

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

Now, we define the residual function over the interval [x0, xf] as follows

R=x0xfR1(x,a0,a1,,an)2dx.(5)

3 Orthonormal Bernstein polynomials of arbitrary order

In this section, OBP is constructed. For arbitrary n, Eq. (6) can be obtained by induction. Now, let n = 3, for simplicity Bi = Bi(x) (i = 0, 1, 2, 3) and OBi = OBi(x), hence

ψ0=B0,OB0=ψ0ψ0,ψ1=B1<B1,OB0>OB0,OB1=ψ1ψ1,ψ2=B2<B2,OB0>OB0<B2,OB1>OB1,OB2=ψ2ψ2,ψ3=B3<B3,OB0>OB0<B3,OB1>OB1<B3,OB2>OB2,OB3=ψ3ψ3,(6)

where OBi (i ≥ 0) can be obtained from Gram-Schmidt orthonormalization technique

ψm=Bm<Bm,OB0>OB0<Bm,OB1>OB1<Bm,OB2>OB2<Bm,OBm1>OBm1,OBm=ψmψm,(for arbitrary m),

and

ψ02=<ψ0,ψ0>=01ψ0ψ0dx,

hence

OB0=8(12t)37,OB1=(13(24t(12t)2+24(12t)3))5,OB2=(56(24t2(12t)+48t(12t)2(485)(12t)3))3,OB3=12t3+36t2(12t)+16t(12t)28(12t)3,

then the following OBP is considered as

y(x)=i=0narOBi(x),x[0,1],(7)

by substituting Eq. (7) in Eq. (4), we can continue the stated algorithm for solving NTBVP.

4 Numerical examples

Now, some numerical examples of NTBVPs are stated to illustrate BCM. All results are obtained by utilizing Maple 14.

Example 1

First, the following NTBVP is considered (see [4])

y=cosysiny+2y+cos(1x2)sin(2x)2(x21)+2,x[1,1],y(1)=0,y(1)=0,yexact=x21,

Applying the proposed technique with n = 4, one may have

yapprox=8.807584016×10309(1x)4(0.250000000000226(x+1))(1x)30.500000000000764(x+1)2(1x)20.249999999994255(x+1)3(1x),

where the obtained error is zero. The graphs of approximated and exact solution y are plotted in Figure 1.

The graphs of approximated and exact solution y for Example 1
Figure 1

The graphs of approximated and exact solution y for Example 1

Example 2

Second, one may solve the Example 1 by OBP. Let n = 3, we can obtain easily the following results (see Table 2). these results can be obtained by OBP method.

Table 1

The error of proposed method for Example 1

Table 2

The error of OBP method for Example 2

y(x)=1+4×109x+x2+2×109x3,x[0,1].

Example 3

Third, the following NTBVP is considered (see [4])

y=sinycosy+cosx(4x21)sin(112x2)+24x,x[12,12],y(12)y(12)=2,y(12)+y(12)=2,yexact=x(4x21),

Applying the proposed technique with n = 4, one may have

yapprox=2.22044604925031×1016(12x)4+(2(x+12))(12x)31.33226762955019×1015(x+12)2(12x)22(x+12)3(12x).

where the obtained error is stated in Table. 3 for n = 3 and n = 4. This shows that the error should decay as n increases. The graphs of approximated and exact solution y are plotted in Figure 2.

The graphs of approximated and exact solution y for Example 3
Figure 2

The graphs of approximated and exact solution y for Example 3

Table 3

The error of proposed method with n = 3, 4 for Example 3

5 Conclusions

This paper deals with the approximate solution of NTBVPs via BCM and OBP. The unity of the function of orthogonality for Bernstein polynomials (BPs) and the simplicity of applying BPs are great merits that make the technique very attractive and easy to use. Illustrative examples are solved to express the validity and applicability of the proposed methods. The approximate solution is coming very close to the exact solution.

Acknowledgement

The authors would like to thank the anonymous reviewers of this paper for their careful reading, constructive comments and nice suggestions, which have improved the paper very much. The Second author wishes to acknowledge financial support from the University of Venda.

References

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About the article

Received: 2017-12-02

Accepted: 2018-03-24

Published Online: 2018-06-26


Conflict of InterestConflict of Interests: The author declares that there is no conflict of interest in the use of the above mentioned software.


Citation Information: Open Physics, Volume 16, Issue 1, Pages 371–374, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0050.

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© 2018 Fateme Ghomanjani and Stanford Shateyi. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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