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

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 inMaple andMatlab.


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][3][4][5], the Adomian decomposition method [6,7] and the linearizability criteria [8,9], finite difference method [10], and Adomian-Pade technique [11].
In order to find solutions of the system (1), with the mixed conditions (2), we can use the collocation points defined by

Fundamental Matrix Relations
The Lucas polynomials   () can be written in the matrix form as where and if  is odd, 2 If  is even, 2 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 X() and its derivatives X () () is where and B 0 = I (+1)×(+1) is the unit matrix.By using the relations ( 14) and ( 15), we obtain the following relations: , () = X () B  D  A  ,  = 0, 1, . . ., ,  = 1, 2, . . ., .

Method for Solution
Firstly, we can write the system (1) in the matrix form where 2, () . . .
The fundamental matrix equation (28) corresponding to (1) can be written in the form This is a linear system of ( + 1) algebraic equations in the ( + 1) unknown Lucas coefficients such that ,  = 1, 2, . . .,  ( + 1) . ( 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 W is singular, then rows of the matrix (33) can be replaced with any rows of the matrix (29).
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  = 6 are obtained 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 2 and Figures 2(a) and 2(b) display that the actual and estimated errors are very close to zero and almost identical.
Table 3 and Figures 3(a) and 3(b) show that when the value of  increases, the accuracy of solution increases.
Table 4 and Figures 4(a) and 4(b) show that the value of  is increased; the actual absolute errors decrease rapidly.
In addition, this problem was solved by Akyüz-Das ¸oglu 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.(62) It is seen from Table 6 and Figures 5(a), 5(b), and 5(c) that the accuracy increases as the  increase.
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  = 9 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.

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.
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 | ,, ()|.

Table 2 :
Comparison of the actual and estimated absolute errors for  = 6 and  = 8, 11 of the problem (52).

Table 6 :
Numerical results of the exact solutions and the approximate solutions for  = 2, 5, 9 of problem (60).