A Block Procedure with Linear Multi-Step Methods Using Legendre Polynomials for Solving ODEs

In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The obtained numerical results reveal that the proposed method is efficient.


Introduction
Numerous problems in Physics, Chemistry, Biology and Engineering science are modeled mathematically by ordinary differential equations (ODEs), e.g., series circuits, mechanical systems with several springs attached in series lead to a system of differential equations [1].Since most realistic differential equations do not have exact analytic solutions, therefore, approximate and numerical techniques [2]- [14] are used extensively.Recently introduced power series method [15] has been used for solving a wide range of problems.This method has proven rather successful in dealing with both linear as well as nonlinear problems, as it yields analytical solutions and offers certain advantages over standard numerical methods.It is free from rounding off errors since it does not involve discretization, and is computationally inexpensive.Ercan and Mustafa [16] have applied this method to a system of differential-algebraic equations.
In this paper, we present an efficient numerical method to solve numerically the ODEs.The proposed method is a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3) [17] using Legendre polynomials as the basis functions.In addition, we give discrete methods used in block and implement it for solving the non-stiff IVPs that were the continuous interpolant derived and collocated at grid and off-grid points.In this article, we consider the general form of the first order initial value problem The plan of the paper is as follows: In Section 2, the derivation of the proposed methods is presented.In Section 3, stability and convergence analysis of the block schemes is given.In Section 4, two numerical examples are considered.The paper ends with summary and conclusions in Section 5.

The Derivation of the Proposed Methods
In this section, we derive discrete methods to solve (1)  , for some positive number N. The proposed method depends on the perturbed collocation method with respect to the power series method is used as the basis for collocation approximation with the Legendre polynomials as the perturbation term.
In the first we consider the approximate solution of the perturbed form of (1) in the following power series [18]- [21] ( ) ( ) where Substituting from Equation (2) in Equation (1) we have where ( ) and λ is a perturbed parame- ter.In particular, we shall be dealing with cases K = 1, 2 and 3 in ( 2) and ( 3).
The well-known Legendre polynomials are defined on the interval [ ] 1,1 − and can be determined with the aid of the following recurrence formula [22] [23] ( ) ( ) ( ) where the first four polynomials are In order to use these polynomials on the interval [ ] x x + we define the so called shifted Legendre polynomials by introducing the change of variable

Cases Study
Case 1: In this case, we take the polynomial ( ) L x x = and use ( 6), then collocate this equation at n x and 1 n x + , we obtain ( ) In addition, from Equation (3) we can deduce that ( ) Then, the Equation (4) will reduce to the following form ( ) ( ) We now collocate Equation ( 7) at n i x + , 0,1 i = and interpolate (2) at n x x = , we get a system of three Equations with ( ) Using a suitable method to solve the above system obtain ( ) ( ) Now, the required numerical scheme of the proposed method will be obtained if we collocate the above Equation ( 9) at Which is the well-known trapezoidal rule.

Case 2: If K = 2
In this case, we take the polynomial ( ) ( ) x + , we obtain ( ) ( ) ( ) In addition, from Equation (3) we can deduce that ( ) Then, the Equation (4) will reduce to the following form ( ) ( ) We now collocate Equation (11) at n i x + , 0,1, 2 i = and interpolate (2) at n x x = , we get a system of four equations with ( ) .
Using a suitable method to solve the above system to obtain From (2), we have ( ) Now, the required numerical scheme of the proposed method will be obtained if we collocate the above Equation ( 13) at Case 3: If K 3 For case K = 3, we collocate the continuous scheme ( ) at grid and off-grid points

Stability and Convergence Analysis of the Block Schemes
In this section, we present a summary on the order, the error constant and the convergence of the proposed block schemes.This summary in given in the following table. Step

Numerical Examples
In this section, we implement the proposed method with K = 2 and K = 3 to solve two first order initial value problems, and compare the obtained numerical results with those obtained from using the fourth-order Runge-Kutta method (RK4).

Example 1.
Consider the following IVP With the exact solution ( ) e x y x − = .The numerical results of this example are presented in Table 1 and Table 2 with the cases K = 2 and K = 3, respectively.In these tables, we presented a comparison the obtained numerical results using the proposed method with the exact solution and those numerical results obtained from using RK4.
Consider the following IVP ( ) ( ) ( ) With the exact solution ( ) 1 e x y x − = − .The numerical results of this example are presented in Figure 1 and Figure 2 with the cases K = 2 and K = 3, respectively.In these figures, we presented a comparison the obtained numerical results using the proposed method with the exact solution and those numerical results obtained from using RK4.

Summary and Conclusions
In this paper, we presented three new block-schemes (K = 1, K = 2 and K = 3) that are convergent and absolutely stable.We used the proposed method to solve numerically a wide-range of linear initial value problems.The results of the presented examples show that our method was capable for solving such problems of IVPs and generates the convergence analysis, and closed to their exact solutions.This method is very simple and effective for a wide-range of ODEs.All computations are made by Matlab.

Figure 1 .
Figure 1.A comparison the proposed method at K = 2 with the exact solution and RK4: Example 2.

Figure 2 .
Figure 2. A comparison the proposed method at K = 3 with the exact solution and RK4: Example 2.

Table 1 .
A comparison the proposed method at K = 2 with the exact solution and RK4: Example 1.

Table 2 .
A comparison the proposed method at K = 3 with the exact solution and RK4: Example 1.