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Department of Mathematics and Computer Sciences, Umaru Musa Yar'adua University, Katsina, Nigeria

- *Corresponding Author:
- Babangida B

Department of Mathematics and Computer Sciences

Faculty of Natural and Applied Sciences

Umaru Musa Yar'adua University Katsina

Katsina State, Nigeria

**Tel:**+2347067704150

**E-mail:**[email protected]

**Received Date:** October 24, 2016; **Accepted Date:** November 23, 2016; **Published Date:** November 30, 2016

**Citation: **Babangida B, Hamisu M (2016) Convergence of the 2-Point Diagonally
Implicit Super Class of BBDF with Off-Step Points for Solving Stiff IVPs. J Appl
Computat Math 5:330. doi: 10.4172/2168-9679.1000330

**Copyright:** © 2016 Babangida B, 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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

The 2-point Diagonally Implicit Superclass of BBDF with off-step point’s method proposed by Babangida had been studied and further established the necessary conditions for its convergence. Consistency, zero stability and order of the method are discussed.

Off-step; Diagonally implicit 2-point SBBDF; Stiff ODEs; Zero stability; Consistency conditions; Convergence order

Consider the following system of first order n-dimensional system of stiff IVPs of the form:

(1)

with , in the interval a ≤ x ≤ b, where

System (1), which arise frequently in the study of electrical circuits, vibrations, chemical reactions, kinetics, automatic control and combustion, theory of fluid mechanics etc., is said to be stiff if its solution contains components with both slowly and rapidly decaying rates, due to large difference of time scales exhibited by the system [1,2]. Dahlquist [3] defines stiffness as systems containing very fast components as well as very slow components. It is difficult to develop suitable methods for stiff problems. However, considerable research efforts have been made by researchers, such as Suleiman [1,2], Musa [4-8], Abasi [9,10], Ibrahim [11,12], Zawawi [13] and Cash [14], to develop suitable methods for stiff ODEs. This paper discussed the order and convergence of diagonally implicit 2-point superclass of BBDF with off step points for solving (1) developed by Babangida [15] and it is as follows:

The derivation of the method in details can be found in Babangida [15].

This section derives the order of the method (2). Method (2) can be rewritten in matrix form as follows

**Definition 2.1:** The order of the block method (2) and its associated linear operator L is given by

(4)

Expanding the functions and as Taylor series around x and substitute in (4) gives

(5)

**Definition 2.2:** The difference operator (5) and the associated method (2) is considered of order p if E_{0}=E_{1}=E_{2}=…=E_{p}=0 and E_{p+1} ≠ 0.

It follows that

(6)

(7)

(8)

(9)

It was therefore shown from (9) that method (2) is of order 2, with error constant

An acceptable linear multistep method (LMM) must be Convergence. This section shows the convergence of the method (2). Consistency and zero stability are the necessary and sufficient conditions for the convergence of any numerical method. We start by presenting the following definitions related to the convergence of (2) taken from Musa [16] as follows:

**Definition 3.1 (Linear multistep method)**

A general k-step linear multistep method is defined as

(10)

Where *α _{j}* and

**Definition 3.2 (Linear difference operator)**

The linear operator L associated LLM (10) is defined by

(11)

Where y(x) is an arbitrary test function and it is continuously differentiable on [a, b]. Expanding y(x + jh) and ý(x+jh) as Taylor series about x, and collecting common terms yields.

(12)

Where C_{q} are constants given by

**Definition 3.3 (Consistency)**

The LMM (10) is said to be consistent if its order p ≥ 1.

It also follows from (13) that the LMM (10) is consistent if and only if

(14)

**Definition 3.4 (Zero stability)**

A Linear Multistep Method (10) is said to be zero stable if no root of its first characteristics polynomial has modulus greater than one and that any root with modulus one is simple.

**Consistency of the method**

In section 2, it has shown that the 2-point DISBBDF with off-step points proposed by Babangida [15] is of order 2, which satisfies the consistency conditions given in definition 3.3. It now remains to show that the method is consistent.

The method (2) is consistent if and only if the following conditions are satisfied:

(15)

(16)

Where D_{j's} and G_{j's} are previously defined in section 2.

Equation (15) then becomes

, (17)

Hence condition (15) is therefore met.

Equation (16) also becomes

, (18)

(19)

(20)

Thus, the second conditions in (18) and (19) are also satisfied.

The consistency conditions are therefore met. Hence, method (2) is consistent.

It now remains to show that the method is zero stable. The stability polynomial of the method (2) is given by

(21)

For zero stability, we set =0 in (21). Hence we have:

(22)

Solving equation (22) for t gives the following roots:

t=0, t=0, t=0.350014 and t=1. (23)

From the definition 3.4, method (2) is zero-stable. Since the method (2) is both consistent and zero stable, it thus method (2) converges.

The 2-point Diagonally Implicit Superclass of BBDF with off-step points method proposed by Babangida had been studied. The necessary conditions for the convergence of method is discussed. The method is zero stable and consistent. It is therefore conclude the method converges.

- Suleiman MB, Musa H, Ismail F, Senu N, Ibrahim ZB (2014) A new super class of block backward differentiation formulas for stiff ODEs. Asian-European journal of mathematics 7: 1-17.
- Suleiman MB, Musa H, Ismail F, Senu N (2013) International Journal of Computer Mathematics 90: 2391-2408.
- Dahlquist G (1974) Problems related to the numerical treatment of stiff differential equations. International Computing Symposium pp: 307-314.
- Musa H, Suleiman MB, Ismail F, Senu N, Ibrahim ZB (2013) An accurate block solver for stiff IVPs. ISRN Applied mathematics. Hindawi.
- Musa H, Suleiman MB, Ismail F (2015) An implicit 2-point block extended backward differentiation formulas for solving stiff IVPs. Malaysian Journal mathematical Sciences 9: 33-51.
- Musa H, Suleiman MB, Senu N (2011) A-stable 2-point block extended backward differentiation formulas for solving stiff ODEs. AIP Conference Proceedings 1450: 254-258.
- Musa H, Suleiman MB, Senu N (2012) Fully implicit 3-point block extended backward differentiation formulas for solving stiff IVPs. Applied mathematical Sciences 6: 4211-4228.
- Musa H, Suleiman MB, Ismail F, Senu N, Ibrahim ZB (2013) An improved 2-point block backward differentiation formula for solving stiff initial value problems. AIP Conference Proceedings 1522: 211-220.
- Abasi N, Suleiman MB, Abbasi N, Musa H (2014) 2-point block BDF method with off-step points for solving stiff ODEs. Journal of soft computing and Applications 39: 1-15.
- Abasi N, Suleiman MB, Ismail F, Ibrahim ZB, Musa H, et al. (2014) A new formula of variable step 3-point block BDF method for solving stiff ODEs. Journal of Pure and Applied Mathematics: Advances and Application12: 49-76.
- Ibrahim ZB, Othman KI, Suleiman MB (2007) Implicit r-point block backward differentiation formula for first order stiff ODEs. Applied Mathematics and Computation 186: 558-565.
- Ibrahim ZB, Othman KI, Suleiman MB (2007) Variable step block backward differentiation formula for first order stiff ODEs. Proceedings of the World Congress on Engineering 2: 2-6.
- Zawawi ISM, Ibrahim ZB, Othman KI (2015) Derivation of diagonally implicit block backward differentiation formulas for solving stiff IVPs. Mathematical problems in engineering 2015.
- Cash JR (1980) On the integration of stiff systems of ODEs using extended backward differentiation formulae. Numerical Mathematics 34: 235-246.
- Babangida B, Musa H (2016) Diagonally implicit super class of block backward differentiation formula with off-step points for solving stiff initial value problems. Book of Abstract and Proceedings of International Research Conference on Qualitative Education and Sustainable Development.
- Musa H (2013) The convergence and order of the 2-point improved block backward differentiation formula. IORS Journal of Mathematics 7: 61-67.

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