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A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations



J.G. Oghonyon, S.A. Okunuga and S.A. Bishop
 
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ABSTRACT

Background and Objective: Over the years, block predictor-corrector method has been limited to predicting and correcting methods without further use. Predictor-corrector method possesses other attributes that utilize the Principal Local Truncation Error (PLTE) to design a suitable step size, tolerance level and control error. This study examined a variable-step-size block predictor-corrector method for solving first-order Ordinary Differential Equations (ODEs). Materials and Methods: The combination of Newton’s backward difference interpolation polynomial and numerical integration methods were applied and evaluated at some selected grid points to formulate the block predictor-corrector method. Nevertheless, this process advances to generate the PLTE of the block predictor-corrector method after establishing the order of the method. Results: The numerical results were shown to demonstrate the performance of the variable step-size block predictor-corrector method in solving first-order ODEs. The complete results were incurred with the aid of Mathematica 9 kernel for Microsoft windows (64 bit). Conclusion: Numerical results showed that the variable step-size block predictor-corrector method is more effective and perform better than existing methods in terms of the maximum errors at all tested tolerance levels as well as designing a suitable step size to control error.

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J.G. Oghonyon, S.A. Okunuga and S.A. Bishop, 2017. A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations. Asian Journal of Applied Sciences, 10: 96-101.

DOI: 10.3923/ajaps.2017.96.101

URL: https://scialert.net/abstract/?doi=ajaps.2017.96.101
 
Received: October 10, 2016; Accepted: February 18, 2017; Published: March 15, 2017



INTRODUCTION

Predictor-corrector method is very essential for finding a suitable step-size1. This study is concerned with approximating the solution of Initial Value Problems (IVPs) for first-order ODEs of the form1:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(1)

The aim was to formulate a variable step-size block predictor-corrector method. This technique of continuing in variable step-size predictor-corrector method started with Milne and it is referred to as Milne’s device1-4. Other researchers proposed block predictor-corrector method for computing the solution of ODEs in the simple form of Adams type as sited5-12. Gear’s method known for stiff problems is the Backward Differentiation Formula (BDF) as stated previously13-15. In addition, this study possesses a lot of computational advantages as discussed previously3,4. Predictor-corrector techniques constantly provide two estimates at each step, thirdly, they are an efficient device for error-control adaptation which has been reported previously7. To present this process, a variable step-size block predictor-corrector method apply the explicit Adams-Bashforth K-Step method as a predictor and the implicit Adams-Moulton K-1-Step method as a corrector of the same order1,10,11.

Furthermore, it is speculated by Adesanya et al.5,6, Anake et al.7, Bakoji et al.8, James and Adesanya9 and Voss and Abbas12 that block predictor-corrector method(s) is a faster method than other non-block predictor-corrector method(s) with better results and as such, ensure convergence. Again, Adesanya et al.5,6, Anake et al.7, Bakoji et al.8, James and Adesanya9 and Voss and Abbas12 suggested that solving block predictor-corrector method(s) simultaneously using fixed step size is sufficient enough to guarantee maximum errors, while others proposed Backward Differentiation Formula (BDF) to provide the solution for stiff ODEs. This study is motivated by the fact that block predictor-corrector method can be extended using the variable step-size technique to solve nonstiff and mildly stiff ODEs.

Definition 1: b-block, r-point method. If r denotes the block size and h is the step size, then block size in time is rh. Let m = 0, 1, 2,... represent the block number and let n = mr, then the b-block, r-point method can be written in the following general form in Eq. 2:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(2)

Where:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

AS and BS are r×r coefficients matrices13.

MATERIALS AND METHODS

Formulation of the block predictor-corrector method: Newton’s backward difference formula was used to formulate the block predictor-corrector method.

Suppose f(x) has a continuous kth derivative, tm = t0+mh, fm = f (tm) and backward differences are presented by Eq. 3:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

where, ∇q fm = fm, then:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(3)

where, f(k) (ε) is the kth derivative of f appraised at some point in an interval having t, tm-k+1 and tm. Assume to fix Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and m = n-1, (1.3) yields:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Where:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Subbing the above in:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

to get:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(4)

Whenever the last term in Eq. 4 is disregarded, the left behind will be called k-step Adams-Bashforth formula which is shown in Eq. 5:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(5)

Expressing the backward differences in terms of the values at continuing points by:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Thus, Eq. 5 can be rewritten as:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(6)

Continuing with Eq. 6 to generate the block k-step Adams-Bashforth Formula8.

Similarly, the implicit multistep methods-the Adams-moulton method can be derived by setting m = n in Eq. 3 and putting into:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

we get:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Ignoring the error term, gives the method as Eq. 7:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(7)

Replacing for ∇jfn in terms of fn, fn-1, fn-2,..., yields the form:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(8)

By continuing with Eq. 8, the block k-1-step Adams-moulton method can be generated16.

Practical application of block predictor-corrector method: Assuming P defines the application program of the block predictor, C defines block corrector application program, with E as the evaluation application program of f with respect to given values of its parameter. If Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations is computed from the block predictor, Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations is calculated one time and employ the corrector at one time as well to obtain Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations, this describes the computation as PEC. Further appraisal of Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations succeeded by another application program of the corrector gives Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and thus, denoted by PEC(2). Implementing the application program of the block corrector m many times can be referred to as PEC(m). Since m is constant, Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations is accepted as the computational solution at Xn+k. At this point, the last computational value for fn+k is preferred as Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and this will be further decided whether or not to execute Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations. Assuming this concluding execution is done, the mode is denoted by P(EC)m or P(EC)m E. Eventually the decision clearly impacts the next step of the execution, when both predicted and corrected numerical values for yn+k+1 will rely on whether fn+k is accepted as Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations or Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations. Finally, for a given m, P(EC)m or P(EC)m E mode utilize the corrector the same number of times; only P(EC)m E requires one more evaluation per step than P(EC)m as expressed3,4.

Theorem 1: If the multistep method 2 is convergent for pth order equations, then the order of 2 is at least p16.

Theorem 2: The order of a predictor-corrector method for first order equations must be >1 if it is convergent16.

Theorem 1 and 2 draw the conclusion that the order and convergence of the method hold.

Implementation of block predictor-corrector method: Concurring to Jain et al.17 and Lambert3,4, the implementation in the P(EC)m or P(EC)m E mode becomes substantial for the explicit (predictor) and implicit (corrector) methods if both are separate of like order and this requirement makes it indispensable for the step number of the explicit (predictor) method to be one step higher than that of the implicit (corrector) method. Consequently, the mode P(EC)m or P(EC)m E can be formally examined in Eq. 9 for m = 1, 2,...;

P(EC)m:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(9)

P(EC)mE:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Remarking that as m→∞, the result of evaluating with either of the above mode will slope to those given by the mode of correcting to convergence.

Moreover, predictor and corrector pair based on method 2 can be implemented. The mode P(EC)m or P(EC)mE specified by Eq. 9, where h is the step size. Since the predictor and corrector both have the same order p.

Theorem 3 demonstrates adequate condition for the convergence of P(EC)m or P(EC)mE.

Theorem 3: Let Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations be a sequence of approximations of yn+1 obtained by a PECE method. If:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

(for all y near yn+1 including Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations) where, L satisfies the condition Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations, then the sequence Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations converges to yn+1.

Proof: The numeric solution satisfies the equation:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

The corrector satisfies the equation:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Subtracting these two equations, we obtain:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Applying the Lagrange mean value theorem to arrive at:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

where, Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations. Thus:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Now:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

This means that the conclusion of Theorem 3 holds as seen17.

In cases, where Cp+1, Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations are the computed error constant of the predictor-corrector method, respectively. The following consequence holds.

Proposition: Suppose the predictor method have order p* and the corrector method have order p. Then: If p*>p (or p*<p with m> p-p*), then the predictor-corrector methods possesses the same order and the same PLTE as the corrector.

If p*<p and m = p-p*, then the predictor-corrector method possesses the same order as the corrector, but different PLTE.

If p*<p and m<p-p*-1, then the predictor-corrector method possesses the same order equal to p*+m (thus less than p).

Specifically, it is observed that, suppose the predictor has order p-1 and the corrector has order p, the PEC answers to get a method of order p. Moreover, the P(EC)m or P(EC)mE scheme has always the same order and the same PLTE as discussed3,4.

Combining1-4, Milne’s device stated that it is viable to estimate the principal local truncation error of the explicit and implicit (predictor-corrector) method without estimating higher derivatives of y(x). Assuming that p = *, where p* and p defines the order of the explicit (predictor) and implicit (corrector) methods with the same order. Directly, for a method of order p, the principal local truncation errors can be written as Eq. 10 and 11:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(10)

Also:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

(11)

where, Wn+j and Cn+j are called the predicted and corrected approximations are given by the method of order p while Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and Cp+1 are independent of h.

Neglecting terms of degree p+2 and above, it is easy to make estimates of the principal local truncation error of the method as Eq. 12:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
(12)

Noting the fact that Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations.

However, the estimate of the principal local truncation error (12) is used to determine whether to accept the results of the current step or to reconstruct the step with a smaller step size. The step is accepted based on a test as prescribed by Eq. 1218. Equation 12 is the convergence criteria otherwise called Milne’s estimate for correcting to convergence. Furthermore, Eq. 12 ensures the convergence criterion of the method during the test evaluation18.

Problem tested: Three test problems are employed. These problems are implemented using variable-step-size block predictor-corrector method.

Test problem 1:y' (x)=xyy(0) = 10<x<1 Solution: Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations
Test problem 2:y’(x) = x-yy(0) = 10<x<1 Solution: y(x) = x+e–x-1
Test problem 3:y’(x) = -10xyy(0) = 10<x<10 Solution: Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

RESULTS AND DISCUSSION

The numeric results to demonstrate the performance of the variable step-size block predictor-corrector method in solving first-order ODEs. The complete result supplied were incurred with the aid of Mathematica 9 Kernel for Microsoft windows (64 bit). The nomenclature utilized are listed in Table 1:

VS-SBP-CM: Variable step-size block predictor-corrector Method
TOL: Tolerance level
h: Step size
MTH: Method used
MAXE: The magnitude of the maximum errors of VS-SBP-CM
ES-TM: Error in stormer-cowell method2 for test problem 1 and 2
1BDF: r = 1-point BDF method9 for test problem 3
2BDF: r = 2-point BDF method9 for test problem 3
3BDF: r = 3-point BDF method9 for test problem 3

The process of estimating the maximum errors and determining the tolerance level are defined as follows:

Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Table 1:
Shows the numeric results of problems 1, 2 and 3 using VS-SBP-CM with comparison to existing methods
Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations

Observing the fact that Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations≠ Cp+1 and Pn+j≠Cn+j· Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and Cp+1 are independent of h.

Where, Image for - A Variable-step-size Block Predictor-corrector Method for Ordinary Differential Equations and Cp+1 are the estimates of the principal local truncation error of the predictor and corrector method. Pn+j and Cn+j are called the predicted and corrected approximations are given by the method of order p.

CONCLUSION

Numeric results have demonstrated the VS-SBP-CM is achieved with the aid of the tolerance level. This tolerance level criteria decide whether the result is accepted or repeated. The results likewise establish the performance of the VS-SBP-CM is remarked to be quicker than the block Stormer-Cowell method and block backward differentiation formula implemented with fixed step size. Hence, it can be resolved that the method formulated is worthy for solving non-stiff and stiff ODEs.

SIGNIFICANCE STATEMENTS

The significance statements of this study are to:

Extend the block predictor-corrector method
Introduce the tolerance level otherwise referred to as convergence criteria
Design a suitable step size
Control the error with the aid of a suitable step size

ACKNOWLEDGMENT

The authors would like to thank Covenant University for providing financial support through grants during the study period.

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