In recent times, the integration of Ordinary Differential Equations (ODEs) is carried out using some kinds of block methods. In this paper, we propose an order six block integrator for the solution of first-order ODEs of the form:

(1)

Where f is continuous within the interval of integration [a,b]. We assume that f satisfies Lipchitz condition which guarantees the existence and uniqueness of solution of eqn. (1). For the discrete solution of (1) by linear multi-step method has being studied by authors like [1] and continuous solution of eqn. (1) and [2-4]. One important advantage of the continuous over discrete approach is the ability to provide discrete schemes for simultaneous integration. These discrete schemes can be reformulated as general linear methods (GLM) [5]. The block methods are self-starting and can be applied to both stiff and non-stiff initial value problem in differential equations. More recently, authors like [6-10] and to mention few, these authors proposed methods ranging from predictor- corrector to hybrid block method for initial value problem in ordinary differential equation.

In this work, hybrid blocks method with two off-grid using Power series expansion [11,12]. This would help in coming up with a more computationally reliable integrator that could solve first order differential equations problems of the form eqn. (1).

In this section, we intend to construct the proposed two-step LMMs which will be used to generate the method. We consider the power series polynomial of the form:

(2)

which is used as our basis to produce an approximate solution to (1.0) as

(3)

And

(4)

where a_j are the parameters to be determined, m and t are the points of collocation and interpolation respectively. This process leads to (m+t-1) of non-linear system of equations with (m+t-1) unknown coefficients, which are to be determined by the use of Maple 17 Mathematical software.

Using eqns. (3) and (4), m=1 and t=5 our choice of degree of polynomial is (m+t-1). Eqn. (3) is interpolates at the point x=x_n and eqn. (4) is collocated at which lead to system of equation of the form

(5)

(6)

With the mathematical software, we obtain the continuous formulation of eqns. (5) and (6) of the form

(7)

After obtaining the values of α_j and β_j,j=0 and in eqn. (7)

We evaluated at the point which gives the following set of discrete schemes to form our hybrid block method.

(8)

Eqn. (8.0) are of uniform order 5, with error constant as follows

Consistency

Definition: The Linear Multistep method is said to be consistent if it is of order

P≥ 1 and its first and second characteristic polynomial defined as and where Z satisfies

See Lambart (1973).

The discrete Schemes derived are all of order than one and satisfy the condition (i)-(iii).

The block method is defined by Fatunla (1988) as

where

and are chosen r x r matrix coefficient and m=0,1,2… represents the block number, n=mγ the first step number in the m-th block and r is the proposed block size.

The block method is said to be zero stable if the roots of R_j,j=1(1)k of the first characteristics polynomial is

satisfies |Rj|≤ 1, if one of the roots is +1, then the root is called Principal Root of ρ(R).

Where

and

The first characteristics polynomial of the scheme is

λ³(λ-1)=0

λ₁=λ₂=λ₃=0 or λ₄=1

We can see clearly that no root has modulus greater than one (i.e λ_i≤1))∀i . The hybrid block method is zero stable.

Problem 1

y'= y, y (0) =1, h = 0.1

Exact solution

y(x)=exp(x) (Table 1).

Table 1: Comparison of approximate solution of problem 1.

Problem 2

y'= 0.5(1− y), y (0) = 0.5, h = 0.1

Exact solution

y(x)=1-0.5e-0.5x (Table 2).

Table 2: Comparison of approximate solution of problem 2.

We observed that from the two problems tested with this proposed block hybrid method the results converges to exact solutions and also compared favorably with the existing similar methods (see Tables 1 and 2).

In this paper, we have presented Hybrid block method algorithm for the solution of first order ordinary differential equations. The approximate solution adopted in this research produced a block method with stability region. This made it to perform well on problems. The block method proposed was found to be zero-stable, consistent and convergent.

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