Generalized One-Step Third Derivative Implicit Hybrid Block Method for the Direct Solution of Second Order Ordinary Diﬀerential Equation

In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary diﬀerential equations using collocation and interpolation approach. To derive this method, the approximate solution power series is interpolated at the ﬁrst and oﬀ-step points and its second and third derivatives are collocated at all points in the given interval. Besides having good numerical method properties, the new developed method is also superior to the existing methods in terms of accuracy when solving the same problems.


Introduction
This article proposes a general one-step third derivative implicit hybrid block method (GOHBM) for the direct solution of the second order ODEs in the form y = f (x, y, y ), y(a) = y 0 , y 0 (a) = y 0 , a x b (1) with the assumption that f is differentiable and satisfies Lipchitz's condition which guarantees the existence and uniqueness of the solution ( [10]).
Block methods which are widely used by many scholars for solving (1) were first introduced by [14] and later by [9] mainly to provide starting values for predictor-corrector algorithms. Those methods produced better accuracy than the usual step by step methods. [8], on the other hand, extended Milne's idea to develop block methods for solving ODEs. In order to obtain higher order methods and hence to increase the accuracy of the approximate solution, [4] proposed hybrid block methods which included off-step point(s) in the derivation of the algorithms. Furthermore, hybrid block methods were used to circumvent Dahlquists barrier conditions which stipulate that the order of a k-step Linear Multistep Method (LMM) cannot exceed k + 1 for k is odd or k + 2 for k is even for the method to be zero-stable ( [6]). In addition, hybrid block methods are also known to share with Runge-Kutta methods their favourable advantage of being self starting and more accurate since they are implemented as a block.
In hybrid block methods, step and off-step points are combined to form a single block for solving ODEs ( see [4], [15], [12]). In addition, [16] introduced second derivative methods which are special types of hybrid methods (referred by [14] as Obrechkoff methods) to enhance the accuracy of the approximation which shown to reach an order k + 2 . Meanwhile, some scholars such as [5], [11] proposed a Simpson's-type second derivative method for the solution of stiff system of first order IVPs. Their work motivated us to propose a new generalized one step third derivative implicit hybrid block method for solving second order ODEs directly using interpolation and collocation in the form where n = 0, 1, 2, ..., N − 1, h = x n − x n−1 is the constant step size for the partition π N of the interval [a, b] which is given by π N = [a = x 0 < x 1 < ... < x N −1 < x N = b], α it , β it and γ it are unknown coefficients, g n+it = f n+it and g n+1 = f n+1 .

Development of the Method
Let us assume the following power series be the approximate solution to (1) where r and s are the number of interpolation and collocation points respectively. Differentiating (2) twice and thrice yields Interpolating (2) at x n+r , r = {0, t} and collocating (3) and (4) at x n+s , s = {0, t, 1} where t ∈ (0, 1), and on combining gives a system of equations in matrix form AX = U where A = a 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 T , U = y n y n+t f n f n+t f n+1 g n g n+t g n+1 T , and Solving (5) for the unknown constant a j s using matrix manipulation and substituting them back into (2) gives a continuous hybrid linear multi-step method in the form γ it g n+it + γ 1 g n+1 ] (6) whose first derivative is Evaluating (6) at x = x n+1 and (7) at x = x n+i , i = {0, t, 1} produces the following general equations in block form where The matrices A (t) , B (i) , D (i) will be described later. To obtain the specific equations of (8), let us consider the following three cases for demonstration. (6) and (7) we get Now, equation (8) can be written as Similarly, replacing t = 1 2 and z = x−x n+ 1 2 h in (6) and (7) produces Thus, equation (8) becomes Finally, putting t = 2 3 and z = x−x n+ 2 3 h in (6) and (7) we have Hence, we can write equation (8) as below

Analysis of the Method
Order of the method The linear operatorL associated with the hybrid block methods formula (8) according to [13] and [7] is said to be of order p if expanding in Taylor series and combining like termŝ where C 0 = C 1 = ... = C p+1 = 0 and C p+2 = 0 The term C p+2 is called the error constant and the local truncation error is given by : For Case (I), substituting t = 1 3 in (9), we get Comparing the coefficients of y i and h i produces C 0 = C 1 = ... = C 7 = 0 with vector of error constants C 8 = For Case (II), substituting t = 1 2 in (9), we have Associating the coefficients of y i and h i yields C 0 = C 1 = ... = C 7 = 0 with vector of error constants C 8 = T which also implies that the order (p) of this method is 6.
For Case (III), substituting t = 2 3 in (9), we get Matching the coefficients of y i and h i yields C 0 = C 1 = ... = C 7 = 0 with vector of error constants C 8 = T which again implies that the order (p) of this method is 6.

Consistency
Definition 3.1. A block method is said to be consistent if its order is greater than one.
We conclude from the three cases above that the order (p) of the hybrid block methods formula (8) is greater than 1 hence the consistency property is satisfied.

Zero Stability
Definition 3.2. The hybrid block method formula (8) is said to be zero stable if no root of the first characteristic equation ρ(R) has modulus greater than one i.e | R s | 1 and if R s = 1 then the multiplicity of R s must not exceed two .
To show that the roots of the first characteristic equation satisfies the prior definition we assume that t ∈ (0, 1) and hence As a result, the developed method is zero stable.

Convergence
Theorem 3.1. (Henrici,1962) Consistency and zero stability are sufficient conditions for a linear multi step method to be convergent The hybrid block method (8) is convergent since it satisfies both the consistency and zero stability conditions.

Numerical Examples
In this section accuracy of the general one-step implicit hybrid block method (8) with order 6 is tested on three experimental problems for the three cases simultaneously, with a fixed step size h = 5 1000 for the first problem h = 1 100 for the second and h = 0.1 32 for the third. The computed results are then compared with recent methods and the new methods is found to have advantages as shown in Tables I-III. Problem (1) : f (x, y, y ) = 3y + 8e 2x , y(0) = 1, y (0) = 1. Exact Solution : y = −4e 2x + 3e 3x + 2 with h = 5 1000 . Source : [2].  (2) : f (x, y, z) = x(y ) 2 , y(0) = 1, y (0) = 1 2 . Exact Solution : y = 1 + ln( 2+x 2−x ) with h = 1 100 . Source : [1]. method, the numerical results suggest that the new method has not only out performed the existing methods, but also circumvent Dahlquists barrier.