Direct Fifth Order Block Backward Differentiation Formulas for Solving Second Order Ordinary Differential Equations

Direct Fifth Order Block Backward Differentiation

Formulas for Solving Second Order Ordinary

Differential Equations

Nooraini Zainuddin [b,c], Zarina Bibi Ibrahim*[a,b], Khairil Iskandar Othman [d] and Mohamed Suleiman [a,b]

[a] Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia. [b] Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor,

Malaysia.

[c] _{Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 32610 Bandar}

Seri Iskandar, Perak, Malaysia.

[d] Department of Mathematics, Faculty of Computer and Mathematical Sciences, Universiti Teknology MARA, 40450 Shah Alam, Selangor, Malaysia.

*Author for correspondence; e-mail: [email protected]

Received: 12 November 2014 Accepted: 18 April 2015

ABSTRACT

In this paper, a new fifth order Direct Block Backward Differentiation Formula (5-DBBDF) for solving second order stiff ordinary differential equations (ODEs) is presented. Five backwards steps are used to generate the coefficients of the 5-DBBDF. This method approximates the solutions at two points concurrently by using fixed step size and solved the second order ODEs directly without reducing the ODEs to a system of first order. Numerical results on some standard problems found in the literature are presented to validate the accuracy of the proposed method. Keywords: block method, second order ODEs, stiff 1. INTRODUCTION This paper focuses on solving directly the general second order ordinary differential equations (ODEs) in the form , , = , (1a) Equation (1a) arises in many fields of applied sciences such as particle tracking, satellite tracking, structural mechanic and network analysis.

Many of these problems cannot be solved analytically and hence the uses of numerical methods are advocated. The majority of numerical methods for solving equation (1a) are by reducing it to a system of first order ODEs and then applying various methods suitable for first order ODEs. Recently, the direct block method for numerical solutions of (1a) have been

extensively proposed in the literature by various researchers. These methods, known as direct block methods were proven to be efficient in providing approximate solution to (1a) with less computational cost and yet accuracy is maintained, see Fatunla [4], Awoyemi [2], Majid [10,11], and Ibrahim [6,7]. [4] proposed zero-stable methods of orders 3/4 for special second order . [2] presented a direct method of solving (1a) based on collocation procedures. Chawla and Sharma [3] developed independently explicit and implicit Runge-Kutta-Nystrom methods for solving (1a). Senu [12] developed an embedded explicit Runge-Kutta-Nystrom (RKN) to integrate special second order differential equations of the form where the solution is oscillatory. [10] have derived a direct two-point block one step method for solving (1a). Although there are many works based on the block method in the literature, the formulas which will be developed have special features as compared to the classical block methods. We seek to obtain a block formula with high accuracy, solved equation (1a) without reduction to first order systems of ODEs and store all the differentiation coefficients in order to prevent repetitive computations of the differentiation coefficients. Apart from aiming at achieving high accuracy, two approximate solutions will be generated simultaneously at each integration step. In the following section, the general theory of block method is summarized briefly. Section 3 discussed the computations of the coefficients of the new method and the implementation of the method using modified Newton Iteration. Finally, in Section 4, numerical tests on second order differential equations which are stiff are performed.

2. GENERAL THEORY ON BLOCK METHODS FOR ODEs

In what follows, we review the theory on k-block r -point methods as discussed by Fatunla

[4] for solving the second order differential equation

(2a)

with initial conditions , where is a known function and and are constants. Commonly, equation (1a) and (2a) are solved by transforming it into an equivalent system of first order differential equations.

Definition: The k-block, r -point methods for (2a) are given by the matrix finite difference

equation

(2b)

where are r by r matrices respectively with elements for

The r-vector and (for is specified as

In this paper, our aim is to extend the theory previously proposed by Ibrahim et al [6]

for stiff (1a) directly. The focus will be on achieving high accuracy with a reduction in the computational cost when solving (1a) directly.

3. DERIVATION OF DIRECT FIFTH ORDER BLOCK BACKWARD DIFFERENTIATION FORMULAS (5-DBBDF)

In this section, we shall derive the coefficients for the 5-DBBDF method for approximating values and in (1a) directly at step and simultaneously. Here, we extend the ideas from [6,11]. We begin by dividing the interval [a,b] into N equally spaced subintervals

such that

The parameter h is called the step size. For 2- point block

method, the interval [a,b] is divided into a series of blocks with each block containing two points

and h is uniform over the interval as shown in Figure 1. From Figure 1, we observed that the

solutions at and are computed using five back values, . Let and denotes the backward difference operator. Define the backward difference representation of the interpolation polynomial as

(3a)

Differentiating Eq. (3a) j times at and substituting leads to

(3b) where . Figure 1. Fifth order 2-Point Block Backward Differentiation Formula.

In order to obtain a recurrence relation for the method of generating function is used. Let (3c) The substitution of into (3c) gives Hence, (3d)

Similarly, the substitution of for into (3c) gives

(3e)

In Table 1, we give the values of for .

Table 1. Values of for at .

m _{0 1 2 3 4 5 6}

0 1

0 0 1 0

The values for and are substituted into equation (3b) to formulate the coefficients of 5-DBBDF. Equation _{is obtained by using values of} and equation is obtained from values .

For the case

Hence, the formula for is as follows,

For the case

The formula for is

The coefficients for and are derived similarly. The point for is substituted into equation (3b).

where .

The values for , given are tabulated in Table 2. The same strategy used for are applied for the formulas at . Given below are the formulas correspond

to .

Finally, we have the formulas for solving (1a) as follows

We now determine the order of the formula given in (3g) by applying the definition of order as discussed by Lambert [9] and Fatunla [4]. The following is the generalization of the linear multistep method for second order ODEs.

where the coefficients are constant and it is assumed that and not all are zero. Consider the linear operator L given as

Table 2. Values of for at .

m

0 1 2 3 4 5 6

0 1

where is an arbitrary function, continuously differentiable on The function may has many higher derivatives, then expanding by Taylor at the point x, we obtain

(3j)

whose coefficients _{are constant of} given as

,

, where .

Definition: The difference operator (3i) associated with linear multistep method in (3h) is said

to be of order p if in (3j),

.

It is shown that this definition can be extended to block methods, see Majid [10]. Thus, the method given by Eq. (3g) is of order five.

3.1 Implementation of the Method

The method derived is fully implicit since the function f is evaluated with unknown

information. We refer the method 5-DBBDF as fully implicit since the formula includes the current point and the future point , therefore predictors are used to evaluate the functions and simultaneously. Since the method is fully implicit, Newton iteration will be implemented, which means that systems of linear equations in the form

must be solved where J is the Jacobian matrix. Therefore, the corresponding linear system to

be solved in matrix form is where

and

and are the back values. Subsequently, the values of and are substituted into the following,

, ,

, ,

where is the number of iterations. The method is implemented in mode where P denotes an application of a predictor, E denotes an evaluation of a function f and C

denotes the corrected values. The sequence of the implementation process takes the following (i) the values of and are computed using the predictor formulas. (ii) apply Newton iterations for and .

(iii) corrected values for and are obtained from the equation (3k).

(iv) apply second stage of the Newton iteration to obtain the corrected values for and .

We use the Euler method to compute the initial starting points .

4. NUMERICAL RESULTS AND DISCUSSION

In this section, we compare the results obtained from the proposed method to Gear’s method of order five (5-GEAR) on some stiff problems taken from Abell and Braselton [1]. Since 5-GEAR is the method for solving stiff first order ODEs, the tested problems will be reduced into the first order form.

Problem 1: , , , .

Eigenvalues: ,

Exact Solution: .

Reduction to First Order ODEs:

, . Exact Solution: , . Problem 2: , , , Eigenvalues: _1,2 125 25 7 , 2 2 i

λ

Reduction to First Order ODEs: , Exact Solution: , The following notations are used in the tables and take the following meaning: h : Step size; user specified AVGE : Average error

MAXE : Maximum error of the computed solution TIME : Computing time obtained in microsecond

5-DBBDF : Direct Fifth Order Block Backward Differentiation Formulas 5-GEAR : Fifth order of Gear’s method

The calculation of errors is given by

where is the t-th component of exact solution and is the t-th component of

computed solution at . When A=1, B=1, equations (4a) corresponds to the mixed error

test, A=1, B=0 corresponds to the absolute error test and when A=0, B=1, equation (4a)

corresponds to the relative error test. For all the test problems, the mixed error test is used. MAXE and AVGE are given as follows, , AVGE=

( )

( )(

)

∑ ∑

where N is the number of equations in the systems and STEP is the total number of integration

steps required. From Tables 3 and 4, it is clearly shown that the 5-DBBDF managed to give competent results for all the step sizes used. It can be seen from these tables, the total number of steps taken by 5-GEAR is doubled of the 5-DBBDF method. This is expected since solving the problem in block method will reduce the total number of steps significantly. Subsequently, the reduction in computational time is predictable. Although the dimension of the Jacobian matrix resulting from the block method is 2 by 2 instead of 1 by 1 for nonblock, this discrepancy is compensated by the number of equations in the first order form which is doubled. Furthermore, the modified Newton iteration applied by 5-DBBDF ensures that the Jacobian is not updated at each integration step. This makes the 5-DBBDF superior in terms of computational time as compared to the reduction method. This advantage already agreed by [5,8,13] and the 5-DBBDF method proved it.

Table 3. Numerical Result for Problem 1.

h METHOD STEP AVGE MAXE TIME

1.00E-02 5-DBDDF 100 2.55257E-05 1.03822E-03 0.000235

5-GEAR 200 5.15796E-03 2.16964E-01 0.000270

1.00E-03 5-DBDDF 1000 8.27473E-07 3.35185E-05 0.000937

5-GEAR 2000 8.38569E-05 3.53427E-03 0.001364

1.00E-04 5-DBDDF 10000 7.43005E-09 2.98213E-07 0.007954

5-GEAR 20000 9.15787E-07 3.56662E-05 0.012132

1.00E-05 5-DBDDF 100000 7.34114E-11 2.93615E-09 0.075016

5-GEAR 200000 9.22906E-09 3.57143E-07 0.115680

1.00E-06 5-DBDDF 1000000 3.02881E-11 6.41417E-11 0.748380

5-GEAR 2000000 9.24319E-11 3.57191E-09 1.153843

Table 4. Numerical Result for Problem 2.

h METHOD STEP AVGE MAXE TIME

1.00E-02 5-DBDDF 100 7.12856E-06 8.87588E-04 0.000201

5-GEAR 200 1.03315E-02 2.56382E-01 0.000226

1.00E-03 5-DBDDF 1000 1.02738E-05 5.87295E-04 0.000993

5-GEAR 2000 5.15329E-04 8.15426E-02 0.001294

1.00E-04 5-DBDDF 10000 1.21799E-07 6.78732E-06 0.007663

5-GEAR 20000 7.36230E-06 1.14202E-03 0.012091

1.00E-05 5-DBDDF 100000 1.23406E-09 6.87484E-08 0.070875

5-GEAR 200000 7.61408E-08 1.18572E-05 0.155613

1.00E-06 5-DBDDF 1000000 1.41522E-11 7.05869E-10 0.706537

5-GEAR 2000000 7.63948E-10 1.19015E-07 1.185999

5. CONCLUSION A block method which approximates the solution of second order ODEs directly by using fixed step size is presented. The numerical test proved that the 5-DBBDF can be an alternative solver for second order ODEs. The comparison of the numerical results obtained by 5-DBBDF with 5-GEAR demonstrates the reliability of the proposed method. ACKNOWLEDGEMENTS The authors wish to thank the anonymous referees for their careful reading of the manuscript. This work is financially supported by Institute for Mathematical Research, Universiti Putra Malaysia under Fundamental Research Grant Scheme (project code 01-04-10-888FR).

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