Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 625984, 14 pages
http://dx.doi.org/10.1155/2015/625984

Research Article

Lucas Polynomial Approach for System of High-Order Linear Differential Equations and Residual Error Estimation

Muhammed Çetin,¹ Mehmet Sezer,¹ and Coşkun Güler²

¹Department of Mathematics, Faculty of Science, Celal Bayar University, Manisa, Turkey
²Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yıldız Technical University, Istanbul, Turkey

Received 15 November 2014; Accepted 5 January 2015

Academic Editor: Mingshu Peng

Copyright © 2015 Muhammed Çetin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An approximation method based on Lucas polynomials is presented for the solution of the system of high-order linear differential equations with variable coefficients under the mixed conditions. This method transforms the system of ordinary differential equations (ODEs) to the linear algebraic equations system by expanding the approximate solutions in terms of the Lucas polynomials with unknown coefficients and by using the matrix operations and collocation points. In addition, the error analysis based on residual function is developed for present method. To demonstrate the efficiency and accuracy of the method, numerical examples are given with the help of computer programmes written in Maple and Matlab.

1. Introduction

The systems of differential equations with variable coefficients have been encountered in many scientific and technological problems. Some of these differential equation systems do not have analytic solutions, so numerical methods are required. The systems of linear differential equations have been solved by many mathematicians and engineers by using the various methods such as variational iteration method [1], the differential transform method [2–5], the Adomian decomposition method [6, 7] and the linearizability criteria [8, 9], finite difference method [10], and Adomian-Pade technique [11].

Taylor, Chebyshev, Legendre, Berstein, Hermite, Laguerre, and Bessel matrix methods are used for solving differential and integral equations, integrodifferential-difference equations, and their systems in [12–20]. In this paper, by means of the above-mentioned methods and the Lucas polynomials, we have developed a new method called Lucas collocation method to solve the system of high-order linear differential equations with variable coefficients in the form

under the mixed conditions

where is an unknown function, and are the known continuous functions defined on interval , and coefficients , and are the real constants.

In addition, by improving the present method with the help of the residual error function used in [21–25], we obtain the corrected approximate solutions of the system (1) expressed in the truncated Lucas series form

where

is the Lucas polynomial solution and

is the solution of the error problem obtained with the aid of the residual error function. Here ,   and ,   are the unknown Lucas coefficients, and ,   are the Lucas polynomials defined by

[26–28]. The purpose of this study is to improve the approximate solutions for high-order systems of ODEs by means of the residual error function and to give an efficient and useful error estimation via the error problem.

In order to find solutions of the system (1), with the mixed conditions (2), we can use the collocation points defined by

2. Fundamental Matrix Relations

The Lucas polynomials can be written in the matrix form as

where

and if is odd,

If is even,

We can write the approximate solutions given by (4) in the matrix form

where

From (8) and (12), we obtain the matrix relation

Also, the relation between the matrix and its derivatives is

where

and is the unit matrix.

By using the relations (14) and (15), we obtain the following relations:

Hence, we can write the matrix relations as

where

3. Method for Solution

Firstly, we can write the system (1) in the matrix form

where

By substituting the collocation points (7) into (20) we obtain the system of matrix equations

or the compact form

where

From the relation (18) and the collocation points (7), we have

or, briefly,

where

By substituting (26) into (23), we obtain the fundamental matrix equation as

In (28) the full dimensions of the matrices , , , , , and are , , , , , and , respectively.

The fundamental matrix equation (28) corresponding to (1) can be written in the form

This is a linear system of algebraic equations in the unknown Lucas coefficients such that

By using the conditions (2) and the relations (18), the matrix form for the conditions is obtained as

where

Hence, the fundamental matrix form for conditions is

such that

Consequently, by replacing the row matrices (33) by last rows of the matrix (29), we obtain the new augmented matrix

We do not have to change the last rows of the matrix equation given by (29). If the matrix is singular, then rows of the matrix (33) can be replaced with any rows of the matrix (29). If , then we can write

Hence, the unknown Lucas coefficients matrix is determined. We can find the Lucas polynomial solutions

4. Residual Correction and Error Estimation

In this section, we will give an error estimation for the Lucas polynomial solutions (4) with the residual error function [21–25]. Moreover, we will improve the solution (4) by means of the residual error function. Firstly, we can define the residual function of the method as

Here, represent the Lucas polynomial solutions given by (4) of the problem (1) and (2). Hence, satisfies the problem

Also, the error function can be defined as

where are the exact solutions of the problem (1) and (2). From (1), (2), (38), and (40), we obtain the error differential equation system

with the homogeneous mixed conditions

or, openly, the error problem

Here, note that the nonhomogeneous mixed conditions

are reduced to homogeneous mixed conditions

The error problem (43) can be solved by using the procedure in Section 3. Thus, we obtain the approximation to as follows:

Consequently, the corrected Lucas polynomial solution is obtained by means of the polynomials and . Also, we construct the error function , the estimated error function , and the corrected error function .

5. Numerical Examples

In this section, the several numerical examples are given to demonstrate the efficiency and applicability of our method. The computations related to the examples are calculated by using a computer programme which is called Maple and the figures are drawn in Matlab. In tables and figures, we calculate the values of the Lucas polynomial solution , the corrected Lucas polynomial solution , the actual absolute error function , and the estimated absolute error function .

Example 1. Let us consider the system of second-order linear differential equations given by

with the boundary conditions

which has the exact solutions and [30]. In this problem , , , , , , , , , , , , , , , and .
The approximate solutions and for are given by

The set of the collocation points given by (7) for , , and is calculated as

From (28), the fundamental matrix equation of the problem (47) is written as

By applying the procedure in Section 3, we obtain the Lucas polynomial solutions for as and , which are the exact solutions.

Example 2. Let us consider the linear differential equations system given by

with the initial conditions

which has the exact solutions and [12, 29]. In this problem , , , , , , , , , , , and .
We can write the fundamental matrix equation of the problem (52) from (28) as

By using our method, the approximate solutions of the problem (52) for are obtained as

In order to calculate the corrected Lucas polynomial solutions, let us consider the error problem

such that , , and the residual functions are

By solving the error problem (56) for , the estimated Lucas error functions and to and are obtained as

Thus, we can calculate the corrected Lucas polynomial solutions and as

It is seen from Table 1 and Figures 1(a) and 1(b) that the accuracy of solution increases when the values of and increase.

Table 1: Numerical results of the exact solutions and the approximate solutions for , and of problem (52).

Exact solution
Lucas polynomial solutions

0.1
0.806485719482
0.806485689143
0.806485719287
0.806485719292

0.2
0.625251708173
0.625251667103
0.625251707990
0.625251708032

0.5
0.145975834384
0.145975783291
0.145975834235
0.145975834133

0.8
−0.252886020789
−0.252886103976
−0.252886020930
−0.252886021210

1.0
−0.482526627107
−0.482526066408
−0.482526628764
−0.482526626284

0.1
0.098405441705
0.098405456828
0.098405441810
0.098405441788

0.2
0.193895101008
0.193895121487
0.193895101121
0.193895101070

0.5
0.466457257480
0.466457282978
0.466457257577
0.466457257606

0.8
0.724228025488
0.724228067006
0.724228025625
0.724228025747

1.0
0.890857245275
0.890856966173
0.890857246295
0.890857244962

(a)

(b)

Figure 1: (a) Comparison of the exact solution and the approximate solutions and . (b) Comparison of the exact solution and the approximate solutions and .

Table 2 and Figures 2(a) and 2(b) display that the actual and estimated errors are very close to zero and almost identical.

Table 2: Comparison of the actual and estimated absolute errors for and of the problem (52).

Actual absolute errors
Estimated absolute errors

0.1

0.2

0.5

0.8

1.0

0.1

0.2

0.5

0.8

1.0

(a)

(b)

Figure 2: (a) Comparison of the actual and estimated absolute errors for . (b) Comparison of the actual and estimated absolute errors for .

Table 3 and Figures 3(a) and 3(b) show that when the value of increases, the accuracy of solution increases.

Table 3: Numerical results of the exact solutions and the Lucas polynomial solutions for of problem (52).

Exact solution
Lucas polynomial solutions

0.1
0.806485719482
0.806485689143
0.806485719432
0.806485719479

0.2
0.625251708173
0.625251667103
0.625251708124
0.625251708169

0.5
0.145975834384
0.145975783291
0.145975834317
0.145975834379

0.8
−0.252886020789
−0.252886103976
−0.252886020880
−0.252886020798

1.0
−0.482526627107
−0.482526066408
−0.482526625954
−0.482526627122

0.1
0.098405441705
0.098405456828
0.098405441730
0.098405441706

0.2
0.193895101008
0.193895121487
0.193895101033
0.193895101010

0.5
0.466457257480
0.466457282978
0.466457257512
0.466457257479

0.8
0.724228025488
0.724228067006
0.724228025534
0.724228025484

1.0
0.890857245275
0.890856966173
0.890857244698
0.890857245264

(a)

(b)

Figure 3: (a) Comparison of the exact solution and the approximate solutions . (b) Comparison of the exact solution and the approximate solutions .

Table 4 and Figures 4(a) and 4(b) show that the value of is increased; the actual absolute errors decrease rapidly.

Table 4: Comparison of the absolute error functions for of problem (52).

Lucas collocation method

0.1

0.2

0.5

0.8

1.0

0.1

0.2

0.5

0.8

1.0

(a)

(b)

Figure 4: (a) Comparison of the absolute error functions . (b) Comparison of the absolute error functions .

In addition, this problem was solved by Akyüz-Daşoğlu and Sezer [12] and Davies and Crann [29]. Now, let us compare our method (LCM) with the other methods (Chebyshev method and Stehfest method) given by [12, 29]. Table 5 indicates this comparison.

Table 5: Comparison of the absolute error functions and for different methods of problem (52).

Chebyshev method in [12]
Stehfest Method in [29]
Present method
Present method

0.1

0.2

0.5

0.8

1.0

Chebyshev method in [12]
Stehfest Method in [29]
Present method
Present method

0.1

0.2

0.5

0.8

1.0

It is seen from Table 5 that the present method gives better approximations than the other methods given by [12, 29].

Example 3. Let us consider the linear differential equations system given by

with the initial conditions

which has the exact solutions , , and [2]. By using the method, the approximate solutions of the problem (60) for are obtained as

It is seen from Table 6 and Figures 5(a), 5(b), and 5(c) that the accuracy increases as the increase.

Table 6: Numerical results of the exact solutions and the approximate solutions for of problem (60).

Exact solution
Lucas polynomial solutions

0.1
1.105170918076
1.106706136074
1.105169808728
1.105170918072

0.2
1.221402758160
1.226824544296
1.221400910855
1.221402758157

0.5
1.648721270700
1.667653401853
1.648719504255
1.648721270696

0.8
2.225540928492
2.229192708743
2.225540191066
2.225540928487

1.0
2.718281828459
2.670613607410
2.718246576634
2.718281828322

0.1
0.099833416646
0.098994748986
0.099833619019
0.099833416647

0.2
0.198669330795
0.195978995944
0.198669599126
0.198669330795

0.5
0.479425538604
0.474868724650
0.479425418757
0.479425538603

0.8
0.717356090899
0.735663935104
0.717355091659
0.717356090898

1.0
0.841470984808
0.899474898600
0.841480130265
0.841470984845

0.1
2.100175083354
2.101927846772
2.100174590140
2.100175083352

0.2
2.201469336001
2.207711387088
2.201468393096
2.201469335999

0.5
2.526303832591
2.548196169301
2.526302577195
2.526303832587

0.8
2.922247637840
2.923382193410
2.922246514015
2.922247637835

1.0
3.258584134327
3.192784677202
3.258568370866
3.258584134265

(a)

(b)

(c)

Figure 5: (a) Comparison of the exact solution and the approximate solutions . (b) Comparison of the exact solution and the approximate solutions . (c) Comparison of the exact solution and the approximate solutions .

Table 7 shows that while the value of is increased, the errors decrease rapidly. Now, we compare the present method with the differential transform method given by [2].

Table 7: Comparison of the actual absolute errors for of problem (60).

0.1

0.2

0.5

0.8

1.0

0.1

0.2

0.5

0.8

1.0

0.1

0.2

0.5

0.8

1.0

It is seen from Table 8 that the present method (LCM) is very effective compared to the differential transform method (DTM) for problem (60).

Table 8: Comparison of the errors of the methods for problem (60).

Differential transform
method in [2]
Present method

0.1

0.2

0.5

0.8

1.0

Differential transform
method in [2]
Present method

0.1

0.2

0.5

0.8

1.0

Differential transform
method in [2]
Present method

0.1

0.2

0.5

0.8

1.0

Figures 6(a), 6(b), and 6(c) display the actual absolute error functions obtained by present method for and the differential transform method. These figures display that the results gained by the present method are better than those obtained by the differential transform method.

(a)

(b)

(c)

Figure 6: (a) Comparison of the absolute error functions for . (b) Comparison of the absolute error functions for . (c) Comparison of the absolute error functions for .

6. Conclusions

It is known that solving the high-order linear differential equations system is usually very difficult analytically. In this case, it is required to approximate solutions. In this paper, a new method based on the Lucas polynomials with the help of the residual error function for solving system of high-order linear differential equations numerically is presented. When the obtained results are investigated in examples, it can be seen that the developed method is very effective compared to the others. Also, it can be seen from the tables and the figures that the accuracy increased when the value of is increased. The approximate solutions are obtained in a short time with computer programmes such as Maple, Mathematica, and Matlab. We have used the Maple and Matlab for computations and graphics, respectively. Additionally, the presented method can be applied to the other system of linear integral and integrodifferential equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

M. Tatari and M. Dehghan, “Improvement of He's variational iteration method for solving systems of differential equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2160–2166, 2009.

M. Thongmoon and S. Pusjuso, “The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 3, pp. 425–431, 2010.

I. H. Abdel-Halim Hassan, “Application to differential transformation method for solving systems of differential equations,” Applied Mathematical Modelling, vol. 32, no. 12, pp. 2552–2559, 2008.

A. Kurnaz and G. Oturanç, “The differential transform approximation for the system of ordinary differential equations,” International Journal of Computer Mathematics, vol. 82, no. 6, pp. 709–719, 2005.

F. Ayaz, “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 547–567, 2004.

H. Jafari and V. Daftardar-Gejji, “Revised Adomian decomposition method for solving systems of ordinary and fractional differential equations,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 598–608, 2006.

J. Biazar, E. Babolian, and R. Islam, “Solution of the system of ordinary differential equations by Adomian decomposition method,” Applied Mathematics and Computation, vol. 147, no. 3, pp. 713–719, 2004.

F. M. Mahomed, I. Naeem, and A. Qadir, “Conditional linearizability criteria for a system of third-order ordinary differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 6, pp. 3404–3412, 2009.

C. W. Soh and F. M. Mahomed, “Linearization criteria for a system of second-order ordinary differential equations,” International Journal of Non-Linear Mechanics, vol. 36, no. 4, pp. 671–677, 2001.

A. Saadatmandi and J. A. Farsangi, “Chebyshev finite difference method for a nonlinear system of second-order boundary value problems,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 586–591, 2007.

M. Dehghan, M. Shakourifar, and A. Hamidi, “The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique,” Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2509–2521, 2009.

A. Akyüz-Daşoğlu and M. Sezer, “Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 237–247, 2003.

O. R. Işık, Z. Güney, and M. Sezer, “Bernstein series solutions of pantograph equations using polynomial interpolation,” Journal of Difference Equations and Applications, vol. 18, no. 3, pp. 357–374, 2012.

S. Yalçınbaş, M. Aynıgül, and M. Sezer, “A collocation method using Hermite polynomials for approximate solution of pantograph equations,” Journal of the Franklin Institute, vol. 348, no. 6, pp. 1128–1139, 2011.

Y. Öztürk and M. Gülsu, “A numerical approach for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation,” Applied Mathematical Modelling, vol. 37, no. 5, pp. 2709–2716, 2013.

N. Bildik, A. Konuralp, and S. Yalçinbaş, “Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations,” Computers and Mathematics with Applications, vol. 59, no. 6, pp. 1909–1917, 2010.

Ş. Yüzbaşı, “Laguerre approach for solving pantograph-type Volterra integro-differential equations,” Applied Mathematics and Computation, vol. 232, pp. 1183–1199, 2014.

Ş. Yüzbaşı, “An efficient algorithm for solving multi-pantograph equation systems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 589–603, 2012.

Y. Çenesiz, Y. Keskin, and A. Kurnaz, “The solution of the Bagley-Torvik equation with the generalized Taylor collocation method,” Journal of the Franklin Institute, vol. 347, no. 2, pp. 452–466, 2010.

K. Maleknejad and Y. Mahmoudi, “Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 641–653, 2003.

F. A. Oliveira, “Collocation and residual correction,” Numerische Mathematik, vol. 36, no. 1, pp. 27–31, 1980.

İ. Çelik, “Approximate calculation of eigenvalues with the method of weighted residuals-collocation method,” Applied Mathematics and Computation, vol. 160, no. 2, pp. 401–410, 2005.

İ. Çelik, “Collocation method and residual correction using CHEbyshev series,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 910–920, 2006.

S. Shahmorad, “Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1418–1429, 2005.

Ş. Yüzbaşı and M. Sezer, “An improved Bessel collocation method with a residual error function to solve a class of Lane-Emden differential equations,” Mathematical and Computer Modelling, vol. 57, no. 5-6, pp. 1298–1311, 2013.

A. Constandache, A. Das, and F. Toppan, “Lucas polynomials and a standard Lax representation for the polytropic gas dynamics,” Letters in Mathematical Physics, vol. 60, no. 3, pp. 197–209, 2002.

P. Filipponi and A. F. Horadam, “Second derivative sequences of Fibonacci and Lucas polynomials,” The Fibonacci Quarterly, vol. 31, no. 3, pp. 194–204, 1993.

T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, New York, NY, USA, 2001.

A. Davies and D. Crann, “The solution of differential equations using numerical Laplace transforms,” International Journal of Mathematical Education in Science and Technology, vol. 30, no. 1, pp. 65–79, 1999.

A. Saadatmandi, M. Dehghan, and A. Eftekhari, “Application of He's homotopy perturbation method for non-linear system of second-order boundary value problems,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1912–1922, 2009.