Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms

Article History: Received 30 May 2019 Accepted 16 December 2019 Available 01 July 2020 In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation points is modified and developed for solving a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms, via mixed conditions. The method reduces the solution of the nonlinear equation to the solution of a matrix equation corresponding to system of nonlinear algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples along with an error analysis based on residual function are included to demonstrate the validity and applicability of the proposed method.


Introduction
Nonlinear differential equations and the related initial and boundary value problems play an important role in astrophysics, physics and engineering. In recent years, to solve these problems both analytically and numerically which have applications in various branches of pure and applied sciences, several numerical and analytical methods have been given. But it may not be possible to find the analytical solutions of such problems for all coefficient functions.
In this study, we consider the second-order nonlinear ordinary differential equations with quadratic and cubic terms: Q pqr (x)y (p) (x)y (q) (x)y (r) (x) with the mixed conditions 1 k=0 (a kj y (k) (0) + b kj y (k) (b)) = λ j , j = 0, 1, (2) where P k (x), Q pq (x), Q pqr (x) and g(x) are functions defined on the interval 0 ≤ x ≤ b < ∞; a kj , b kj and λ j are appropriate and real constants; y(x) is an unknown function to be determined [6].

*Corresponding Author
In this study, we develop a new numerical methods to find the approximate solutions of Eq. (1) in the truncated Laguerre series form where a n , n = 0, 1, ..., N, N ≥ 2 are the unknown Laguerre coefficients and L n (x), n = 0, 1, ..., N are the Laguerre functions of first kind defined by

Operational matrix relations
Firstly, let us write Eq. (1) in the form where the linear ordinary differential part the nonlinear quadratic part and the nonlinear cubic part 2.1. Matrix representation of linear ordinary differential part Now, we consider Eq.(1) and find the matrix forms of each term in the equation. So, we convert Laguerre polynomial solution (3) to the matrix form as where

Matrix representation of nonlinear quadratic part
Now, we consider matrix representation of nonlinear quadratic part. So, we define the matrices with related to (7) and (9) (y (0) (x)) 2 = L(x)L(x)A, where

Matrix representation of nonlinear cubic part
Let us consider (8) as So, we define the matrices as

Method of solution
Now, we define the collocation points as We substitute the collocation points (12) into Eq.
(1), we have the system of matrix equations for where and By the other hand, we can write following matrix forms of the nonlinear quadratic and nonlinear cubic parts from (8) and (9) for p, q, r = 0, 1, 2 and where Then the fundamental matrix equation is gained from (5)-(13) Briefly, we can write Eq.(14) as where Moreover, fundamental matrix equation (15) can be written in the augmented matrix form

Matrix representation of the conditions
Let us define the matrix form of the conditions given by (2) can be written as Then, we have or briefly, Consequently, in order to find the Laguerre coefficients a n , (n = 0, 1, ..., N ) related with the approximate solution (3) of the problem (1)-(2), by replacing the 2 row matrices (17) by the last 2 rows (or any 2 rows) of the augmented matrix (16), we obtain new augmented matrix Thence the unknown Laguerre coefficients are calculated by solving (18) [7]- [8]. Therefore, the Laguerre polynomial solution can be acquired as y N (x) = N n=0 a n L n (x).

Error analysis
Definition 1 (Residual function). We define the residual function Then |R N (x α )| is called as the residual function on the interval [0, b].
So, that the upper bound of the mean error R n is Proof. In order to see the proof briefly, we consider the Mean Value Theorem and the definition below.
Step 0. Input initial data: P k (x), Q pq (x), Q pqr (x) and g(x). Determine the mixed conditions.
• Step 1. Set N where N ∈ N.
Construct the matrices L(x), C, L(x), C, L(x), C and G then W, V, Z. • Step 7. Input: the augmented matrix arguments, forward elimination, back substitution. Output: A (Solve the system by Gaussian elimination method).
• Step 8. Put arguments a n in the truncated Laguerre series form.
• Step 9. Output data: the approximate solution y N (x).
• Step 10. Construct y(x) is the exact solution of (1). • Step 11. Stop when R N (x) ≤ 10 −k where k ∈ Z + . Otherwise, increase N and return to Step 1.

Illustrative examples
In this section, some examples will be given to show applicability of our method. All the problems have been calculated and plotted by using Maple18 and MatlabR2014b.

Conclusion
In this study, we introduce a matrix method depending on Laguerre polynomials in order to solve a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms numerically. Furthermore, the error analysis is given to show the accuracy of the method. The present method and its error analysis are applied on some illustrative examples which have been shown by the tables. The method has some significant advantages such as; • The present method has short and concise computing procedure by writing the algorithm in Maple18. • The technique gives an alternative way of solution to the second-order nonlinear ordinary differential equations which varies the other methods in literature. • The present method has sufficient results when N is chosen large enough.