Abstract
In this paper, we show the parallel of Adomian Decomposition Method (ADM) and Lobatto-Runge-Kutta Collocation Method (LRKCM) on first order initial value stiff differential equations. The former method provided closed form solutions while the latter gave approximate solutions. We illustrated these findings in two numerical examples. ADM solutions were in series form while those of LRKCM gave sizeable absolute error. We further visualized our findings in respective plots to show the great potentials of ADM over LRKCM in providing analytical solutions to stiff differential equations.
Keywords: Stiff differential equations, Adomian decomposition method, Lobatto-Runge-Kutta collocation method.
Received: 20 June 2017 / Revised: 13 November 2017 / Accepted: 28 November 2017 / Published: 18 December 2017
Contribution/ Originality
This study contributes in showing the originality of ADM in obtaining exact solution to Stiff differential equations, while LRKCM provided approximate solution whose accuracy depended on step size.
1. INTRODUCTION
The exact solution of a stiff differential equation is typically associated with an exponent that has a large magnitude. It include a term that decay exponentially to zero as the independent variable increases, but whose derivative is much greater in magnitude than the term itself. This class of differential equation, in application, arise from phenomena with widely differing time (independent variable) scales. It is very common as mathematical models in physical and biological sciences. They occur in many fields of engineering science particularly in studies of electrical circuits, vibrations, chemical reactions and so on. There are ubiquitous in weather predictions, astrochemical kinetics, control systems and electronics. In general, it application is wide in industrial areas.
A differential equation
is stiff if the exact solution include a term that decays exponentially to zero as t increases. Suppose such a term is
equation (1) are extremely stable but the numerical ones can be extremely unstable with large step size. Numerical methods currently used in literature for obtaining approximate solutions are implicit Euler’s and Runge-Kutta method see Haier and Wanner [1]. Explicit Euler’s method is also used with unreliable result, finite difference and finite element as well. Others are nth order Taylor’s method, linear multistep method like Adams-Bashforth which is implemented as predictor and Adams-Moulton method which is implemented as corrector. The two methods are regarded as predictor-corrector pair. Direct higher order solver like Runge-kutta and LRKCM are also used, see Butcher [2]; Lie and Norsett [3]. All the aforementioned numerical methods are base on obtaining approximate solutions, an approach that is sometimes expensive when higher accuracy is required. The truncation errors for all methods are large and their absolute errors have sizeable values which may not be useful for practical applications. In this paper, we show the great advantages ADM has over LRKCM when used in obtaining solutions to stiff differential equations. The ADM, in this analysis, gave exact solutions to stiff differential equation (1) in series form; while LRKCM gave approximate solutions with errors that increase as the step-size increased. We depict clearly our findings in respective plotted graphs facilitated by Maple software. The use of LRKCM on equation (1) has also been reported by Yakubu, et al. [4].
2. BRIEF CONCEPT OF LRKCM AND ADM
2.1. LRKCM
Consider the general form stiff differential equation as given in equation (1), y = y(t) with the conditions given as
where
h = equally spaced step size (it can also be a variable)
Assuming that the stiff differential equation (1) has only one solution, αj(t) and hβj(t) are polynomials given as
about t, as it is done in linear multistep method, and collecting powers of h to obtain the LRKCM. The resulting multistep collocation and interpolation matrix affects the efficiency, accuracy and stability property of equation (3).
deriving multistep collocation method as continuous single finite difference formula of non-uniform order six based on Lobatto points. This was recovered from the popular Lobatto IIIA. Equation (3) becomes a special class of multistep collocation method, for details on LRKCM see Yakubu, et al. [4].
2.2. ADM
Considering an abstract stiff differential equation (1), assuming f is analytic near y = y0, (1) becomes the Volterra integral equation
Volterra integral equation
In ADM, y is considered as
And the nonlinear term in
where An are the Adomian polynomial. For more on Adomian polynomials and the concept of ADM see Adomian [5]; Agom and Ogunfiditimi [6] and Agom and Ogunfiditimi [7]; Agom, et al. [8] and Agom, et al. [9] and the references there in. Formally,
3. NUMERICAL ILLUSTRATIONS
In this section we present some numerical results which are adapted from Yakubu, et al. [4]
Example 1
In relation to equations (1) and (2), consider a stiff differential equation with
Applying equations (2) to (5) on (10), as given in Yakubu, et al. [4] the result by LRKCM is stated in Table 1 with step size of 0.1. From Table 1, the Lagranges interpolation polynomial alongside equation (11) gives the plot in Figure 1A.
Table-1. Exact versus LRKCM values of Example 1
t |
Exact value |
LRKCM |
0.1 |
1.024018962351866 |
1.02401896229202 |
0.2 |
1.048582996382734 |
1.04858299626072 |
0.3 |
1.073702928838836 |
1.07370292865234 |
0.4 |
1.099389726731483 |
1.09938972647798 |
0.5 |
1.125654495329782 |
1.12565449500686 |
Table-2. Exact versus LRKCM values of Example 2.
t |
Exact value |
LRKCM |
0.1 |
1.024018962351866 |
1.02401896229202 |
0.2 |
1.048582996382734 |
1.04858299626072 |
0.3 |
1.073702928838836 |
1.07370292865234 |
0.4 |
1.099389726731483 |
1.09938972647798 |
0.5 |
1.125654495329782 |
1.12565449500686 |
Applying equations (6) to (9) of ADM concept, we have the Adomian polynomial of the nonlinear term as
For details see Agom, et al. [8] and Agom and Ogunfiditimi [6] and the references there in. Where
and so on. Equations (13) to (16) are the terms of equation (12) which is the exact solution of stiff differential
Example 2
Also in relation to equations (1) and (2), consider
The solution is
Figure-1A. Exact versus LRKCM Solutions of Example1
Figure-1B. Exact versus ADM Solutions of Example1
Following similar steps in example 1, we have Table 2 showing solutions by analytical method and LRKCM and the plot is as given in Figure 2A.
Similarly using ADM concept, we have
Figure: 2B.
Figure-2A. Exact versus LRKCM Solutions of Example2
Figure-2B. Exact versus ADM Solutions of Example2
4. CONCLUSION
In this work, we have been able to show that ADM uses no step-size but provided exact solutions to stiff differential equation (1); whereas, LRKCM when applied to the same class of equation provided solutions that were in agreement (to some extend) with the exact solutions and ADM solutions. These discoveries were illustrated in respective plots and tables. Also, it has been reported in literature that LRKCM has been applied to obtain fairly accurate result with extremely small step-size to this class of equations. However, this comes with computational cost unlike ADM.
| Funding: The study received no formal support. |
| Competing Interests: The authors declares that they are no competing interest. |
| Contributors/Acknowledgement: E. U. Agom conceptualized, designed and carried out the study supervised by F. O. Ogunfiditimi and Edet Valentine Bassey gave information on multistep collocation method. The authors greatly acknowledge the role, thorough and valuable comments by the supervisor (F. O. Ogunfiditimi). |
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