# Hybrid Block Method Algorithms for Solution of First Order Initial Value Problems in Ordinary Differential Equations

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**Corresponding Author:**Ajileye G, Department of Mathematics and Statistics, Federal University Wukari, Wukari, Taraba State, Nigeria, Tel: 2348034906427, Email: [email protected]

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Received Date: Apr 16, 2018 /
Accepted Date: May 30, 2018 /
Published Date: Jun 19, 2018 *

**Keywords:**
Collocation; Interpolation; Linear multistep method; Hybrid and power series polynomial

#### Introduction

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).

#### Derivation of Hybrid Method

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.

#### Hybrid Block Method

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).

#### Zero Stability of the block Method

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

λ^{3}(λ-1)=0

λ_{1}=λ_{2}=λ_{3}=0 or λ_{4}=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.

#### Numerical Examples

**Problem 1**

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

**Exact solution**

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

x | Exact Solution | Scheme | Error in Scheme | Error [2] |
---|---|---|---|---|

0.1 | 1.105170918075648 | 1.105170917860730 | 2.149179E-10 | 1.226221039551945e-05 |

0.2 | 1.221402758160170 | 1.221402757685120 | 4.7505E-10 | 1.355183832019158e-05 |

0.3 | 1.349858807576003 | 1.349858806788490 | 7.875129E-10 | 1.497709759790133e-05 |

0.4 | 1.491824697641270 | 1.491824696480820 | 1.16045E-09 | 1.655225270247307e-05 |

0.5 | 1.648721270700128 | 1.648721269097010 | 1.603118E-09 | 1.829306831546695e-05 |

0.6 | 1.822118800390509 | 1.822118798264440 | 2.126069E-09 | 2.021696710463594e-05 |

0.7 | 2.013752707470477 | 2.013752704729200 | 2.741277E-09 | 2.234320409577606e-05 |

0.8 | 2.225540928492468 | 2.225540925030090 | 5.989459E-09 | 2.469305938346267e-05 |

0.9 | 2.459603111156950 | 2.459603106852120 | 4.30483-09 | 2.729005110868599e-05 |

1.0 | 2.718281828459046 | 2.718281824122030 | 4.337016E-09 | 3.01601708376864e-05 |

**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)**.

x | Exact Solution | Scheme | Error in Scheme | Error [7] |
---|---|---|---|---|

0.1 | 0.524385287749643 | 0.524385287750861 | 1.218026E-13 | 5.574430e-012 |

0.2 | 0.547581290982020 | 0.547581290981880 | 1.399991E-13 | 3.946177e-012 |

0.3 | 0.569646011787471 | 0.569646011786286 | 1.184941E-12 | 8.183232e-012 |

0.4 | 0.590634623461009 | 0.590634623462548 | 1.538991E-12 | 3.436118e-011 |

0.5 | 0.610599608464297 | 0.610599608463187 | 1.110001E-12 | 1.929743e-010 |

0.6 | 0.629590889659141 | 0.629590889658614 | 5.270229E-12 | 1.879040e-010 |

0.7 | 0.647655955140643 | 0.647655955142752 | 2.10898E-12 | 1.776835e-010 |

0.8 | 0.664839976982180 | 0.664839976969201 | 1.297895E-11 | 1.724676e-010 |

0.9 | 0.681185924189113 | 0.681185924158290 | 3.08229E-11 | 1.847545e-010 |

1.0 | 0.696734670143683 | 0.696734670139561 | 4.121925E-11 | 3.005770e-010 |

**Table 2:** Comparison of approximate solution of problem 2.

#### Discussion of Result

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**).

#### Conclusion

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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Citation: Ajileye G, Amoo SA, Ogwumu OD (2018) Hybrid Block Method Algorithms for Solution of First Order Initial Value Problems in Ordinary Differential Equations. J Appl Computat Math 7: 390. DOI: 10.4172/2168-9679.1000390

Copyright: © 2018 Ajileye G, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.