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.
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Table 2 and Figures 2(a) and 2(b) display that the actual and estimated errors are very close to zero and almost identical.
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Table 3 and Figures 3(a) and 3(b) show that when the value of increases, the accuracy of solution increases.
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Table 4 and Figures 4(a) and 4(b) show that the value of is increased; the actual absolute errors decrease rapidly.
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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.
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.
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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].
It is seen from Table 8 that the present method (LCM) is very effective compared to the differential transform method (DTM) for problem (60).
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.
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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.