Pell-Lucas Collocation Method to Solve Second-Order Nonlinear Lane-Emden Type Pantograph Differential Equations

: In this article, we present a collocation method for second-order nonlinear Lane-Emden type pantograph differential equations under intial conditions. According to the method, the solution of the problem is sought depending on the Pell-Lucas polynomials. The Pell-Lucas polynomials are written in matrix form based on the standard bases. Then, the solution form and its the derivatives are also written in matrix forms. Next, a transformation matrix is constituted for the proportion delay of the solution form. By using the matrix form of the solution, the nonlinear term in the equation is also expressed in matrix form. By using the obtained matrix forms and equally spaced collocation points, the problem is turned into an algebraic system of equations. The solution of this system gives the coeﬀicient matrix in the solution form. In addition, the error estimation and the residual improvement technique are also presented. All presented methods are applied to three examples. The results of applications are presented in tables and graphs. In addition, the results are compared with the results of other methods in the literature.

On the other hand, there are studies by using the Pell-Lucas polynomials for many types of the differential equations [18], [49], [63], [64]. Since no study has yet been done by using the Pell-Lucas polynomials for the solutions of the nonlinear Lane-Emden type pantograph differential equation (LETPDE), the Pell-Lucas polynomials are used for the approximate solutions of the nonlinear second-order Lane-Emden type pantograph differential equation (LETPDE) in this study.
In this paper, we consider the second-order nonlinear Lane-Emden type pantograph differential equation (LETPDE) with the initial conditions Here, k ∈ N , y(t) is the unknown function, g(t) is a continuous function [0, L] and γ, β, λ, µ are some suitable constants. L > 0 and t = 0 is the single singular point of the LETPDE. Our aim is to find the approximate solution of the problem (1)- (2) in the form Here, a n (n = 0, 1, ..., N ) are the Pell Lucas coefficients, N is any positive integer and Q n (t) are the Pell Lucas polynomials defined by In this study, we use two important properties of the Pell-Lucas polynomials: the recurrence relation [27], [28] Q n (s) = 2sQ n−1 (s) + Q n−2 (s), n ≥ 2, where Q 0 (s) = 2, Q 1 (s) = 2s , and the recurrence relation for the derivative [27], [28] where For more features on the Pell-Lucas polynomials, please see [27], [28]. The rest of this paper is organized as follows: The matrix form of the approximate solution and the required matrix relations are established in Section 2. The Pell-Lucas collocation method is constituted in Section 3. The error estimation are presented in Section 4. The applications of the method are given and the results are discussed in Section 5. In Section 6, the conclusions of the paper are given.

Basic Matrix Relations
In this section, we express the matrix forms of the problem (1)-(2) via the Pell-Lucas polynomials.

Lemma 2.1
The vector Q N (t) can be written as where and if N is even Proof By multiplying the vector T N (t) by the matrix D N from the right side, the vector

Lemma 2.2 The approximate solution based on the Pell-Lucas polynomials in (3) can be expressed in the form
where Proof By multiplying the vector Q N (t) = T N (t)D N by the vector A N from the right, the relation (6) is found. ◻

Lemma 2.3
The matrix relations for the first and second derivatives of the solution form (6) are respectively as follows where Proof When the first and second derivatives of (6) are taken, we have respectively Next, the first and second derivatives of T N (t) are taken to obtain Thus, by writing the relations (9) in place of (8) respectively, we obtain the matrix relations for the first and second derivatives of the solution form (6). where Proof If γt is written instead of t in (7), then it is achieved On the other hand, by multiplying the vector T N (t) by the vector M N (γ) from the right, we gain Finally, the desired result is obtained by substituting the relation (12) in (11). ◻

Lemma 2.5 The matrix representation of the solution (6) for the nonlinear term in (1) is expressed as follows
Proof We can write y k N (t) as Then by substituting (6) in (14), the desired results is obtained. ◻ Lemma 2. 6 The matrix relations of the initial conditions (2) for the assumed solution form (6) are respectively as follows Proof If 0 is written instead of t in (6) and (7), it becomes Consequently, the desired result is found. ◻ (6). In that case, the following matrix relation is obtained

Theorem 2.1 It is assumed that the solution of the problem (1)-(2) is sought in form
Proof y ′′ N (t), y ′ N (t) and y k N (t) in (11) and (13) are substituted in (1) and thus, the proof of the theorem is completed. ◻

Method of the Solution
The purpose of this section is to present the Pell-Lucas collocation method. For this reason, firstly, evenly spaced collocation points are defined. Next, the method is constructed by using these collocation points and the matrix relations in the previous section.

Definition 3.1 The evenly spaced collocation points are defined by
Note here that: According to Theorem 3.1, these collocation points are used in (1). If the point a here becomes 0 , a singularity occurs. In Theorem 3.2, the matrix forms obtained for the conditions are written instead of any two lines in the algebraic equation system obtained in Theorem 3.1. Therefore, in order to prevent the singularity that occurs when a point is 0 , instead of the row consisting of point that cause this singularity in Theorem 3.2, the matrix form formed for the first condition is written.
Theorem 3.1 Suppose that the solution of (1) is of form (6). In this instance, (1) by using the collocation points (17) is reduced to a system of nonlinear algebraic equations as follows where Proof If the collocation points (17) are written in (16), then we get We can briefly write (20) as Here, which completes the proof of the theorem. ◻ (6). Then, the problem (1)-(2) by using the collocation points (17) is reduced to a system of nonlinear algebraic equations as followsW

Theorem 3.2 It is assumed that the solution of the problem (1)-(2) is sought in type
Here, [W N ;G N ] is obtained by writing the matrix forms formed for the conditions instead of any (18).
Proof A new matrix system is created by writing two equations obtained for the conditions (2) in Lemma 2.6 instead of any two rows in the system of algebraic equations in Theorem 3.1. This new system is also represented asW N A N =G N . The important part here is to write the row obtained for the first condition instead of the row with the singularity in the matrix [W N ; Thus, the proof is completed. ◻

Error Estimation
The purpose of this section is to given an error estimation technique with the help of the residual function. In addition, by using this error estimation technique and the Pell-Lucas polynomial solution, the residual improvement technique is also presented.

Theorem 4.1 (Error Estimation) Let y(t) be the exact solution and y N (t) be the Pell-Lucas
polynomial solution with N − th degree of the problem (1)- (2). Then, the error problem can be obtained as follows Now, let's subtract the equations in the problem (24) from equations in the problem (1)-(2), respectively, and thus we have Here, since this expression for i = 0 is y k N (t), the error problem is obtained as Hence, proof of theorem is completed. ◻

Numerical Examples
In and the initial conditions . . .
The exact solution of the problem (28)-(29) is y(t) = 1 + t 4 . The Pell-Lucas polynomial solution of the problem (28)- (29) for N = 3 is sought in the form Here, the colocation points for N = 3 are {t 0 = 0, t 1 = 1 3 , t 2 = 2 3 , t 3 = 1}. Notice that the point t 0 = 0 creates a singularity in (28). For this reason, one of the lines created for the conditions is used instead of this line, which creates a singularity in the continuation of the method. With the help of the Theorem 3.1, the fundamental matrix equation is written as where On the other hand, the matrix representations of the condition (29)  Then, the Pell-Lucas polynomial solution becomes y 3 (t) = 1.5452e + 00t 3 − 6.2262e − 01t 2 − 5.5511e − 17t + 1. Thus, the estimated error function for M = 4 is written as  Table 1. According to Figure 3, a better result is obtained with a larger value of N . According to Figure 4, the estimated absolute error function gives similar results to the actual absolute error function, and the improved absolute error function gives better results than the actual absolute error function. According to Table 1, the results of the present method give better results than the results of the Bernoulli collocation method [3].

Example 5.2
As the next example, let's consider the nonlinear differential equation (34) and the initial conditions . . . .
By using the Theorem 3.1, the fundamental matrix equation is obtained as The actual absolute errors, the estimated absolute errors, and the improved absolute errors of the problem (34)  Their results can also be seen from Table 2. According to all tables and graphs, it is concluded that the method gives very successful results.

Example 5.3
The last example is the nonlinear differential equation and the initial conditions The exact solution of the problem (38)-(39) is y(t) = sint . The solution of the problem (38)- (39) for any N is sought in the form where In Table 3 It can be observed from Figure 11 and Table 3 that a more accurate result is obtained with a larger value of N . The interpretation that the estimated absolute errors are very close to the actual absolute errors and that the improved absolute error function gives better results than the actual error function can be made from Figure 12 and Table 3. It is concluded from all tables and graphs that the presented method is a suitable method for the nonlinear second-order Lane-Emden type pantograph differential equation (LETPDE) (1). . . . .

Conclusions
The aim of this study is to present an effective and a reliable method for a class of the nonlinear differential equations. For this purpose, the Pell-Lucas collocation method is presented in the third part of the article. In addition, in the fourth part of the article, an error estimation method is introduced by using the residual function and with the help of the third part of the article.
Moreover, the residual improved technique is also presented. In the fifth section of the article, the methods presented in the previous sections are tested for three examples. These applications are made by using the Matlab program. Application results are tabulated and graphed. According to these results, it is observed that more accurate results are obtained when the value of N in the method is chosen large enough. In addition, it can be said that the error estimation method is quite successful. The importance of the error estimation method is to have information about the results of the method even when the exact solution of the problem is not known. Another result of the study is that the residual improvement technique also yields appropriate results.
According to all these results, it can be interpreted that the Pell-Lucas collocation method, the error estimation technique and the residual improvement method for the nonlinear second-order Lane-Emden type pantograph differential equation (LETPDE) (1) are quite effective and reliable.
These presented methods can also be applied to other types of the nonlinear differential equations after the necessary adjustments are made.

Declaration of Ethical Standards
The authors declare that the materials and methods used in their study do not require ethical committee and/or legal special permission.