An Adaptive Step-Size Taylor Series Based Method and Application to Nonlinear Biochemical Reaction Model
Essam R. El-Zahar
In this study, a Taylor series based method, which is the multi-step Differential
Transform Method (DTM), is adapted to define a new adaptive step-size semi-analytical
and numerical method. This new approach is applied to a nonlinear biochemical
reaction model to obtain approximate analytical solutions in the form of piecewise
convergent series with easily computable components. The numerical results obtained
by using the present method are compared to those obtained by using the classical
DTM, classical multi-step DTM and Runge-Kutta method. The present method gives
a more accurate approximation with less computational effort than fixed step-size
Received: December 17, 2012;
Accepted: January 29, 2013;
Published: March 21, 2013
Nonlinear differential equations appear in mathematical models for many problems
in applied science and engineering, especially in fluid mechanics, chemical
engineering, biochemistry, solid state physics, plasma waves, optimal control,
etc. Most of these real-life problems are still difficult to solve either analytically
or numerically. Recently, numerical methods, which do not require discretization
of space-time variables, perturbation or linearization, are introduced for finding
analytical solutions of nonlinear problems. Differential Transform Method (DTM)
is one of the most effective, convenient and accurate methods for both weakly
and strongly nonlinear problems. DTM is a semi analytical-numerical method for
solving a wide variety of differential equations and provides the solution in
terms of Taylor series. There are also other methods based on Taylor series
expansion such as Restricted Taylor series method (Gulsu
and Ozis, 2005; Ismail et al., 2005) and
Adomian decomposition method (Sen, 1988; Wazwaz,
1998; Hashim et al., 2006; El-Wakil
and Abdou, 2007; Khalifa et al., 2008; Siddiqui
et al., 2010). But DTM formulizes the Taylor series in a totally
different manner. With this method, the given differential equation and related
boundary conditions are transformed into a recurrence relation which leads to
the solution of a system of algebraic equations as coefficients of a power series
solution and is easily carried out in computer. Because of this property, the
method is no need of linearization of the nonlinear problems and as a result
avoids the large computational works and the round-off errors. Different applications
of DTM can be found by Jang et al. (2000), Koksal
and Herdem (2002), Abdel-Halim Hassan (2002, 2008a,
b), Ayaz (2004a, b),
Arikoglu and Ozkol (2006), Liu
and Song (2007), Momani and Odibat (2007), Chu
and Chen (2008), El-Shahed (2008), Momani
and Erturk (2008), Odibat (2008), Ravi Kanth
and Aruna (2009), Kuo and Lo (2009), Al-Sawalha
and Noorani (2009a, b), Ebaid
(2010), Thongmoon and Pusjuso (2010), Kurulay
and Bayram (2010), Alomari (2011), Demirdag
and Yesilce (2011) Gupta (2011), Biazar
et al. (2012) and Gokdogan et al. (2012a).
Even if DTM is an effective tool for solving several linear and nonlinear problems
arising in science and engineering, it also has some drawbacks. Since DTM gives
a truncated Taylor series solution, it does not exhibit a good approximation
in a large domain. To overcome the shortcoming, the multi-step DTM (MsDTM) was
presented by Odibat et al. (2010), Keimanesh
et al. (2011) Gokdogan et al. (2012b),
Yildirim et al. (2012) and El-Zahar
(2012) and succeeded in obtaining reliable approximate solutions for many
problems. MsDTM is a reliable modification of DTM that provides the solution
in terms of convergent series over a sequence of equal-length sub-intervals.
However, for some important classes of problems and for the sake of accuracy
and efficiency, it is necessary to allow variable-length step-size to be used
(Lopeza et al., 1997; Kobeissia
and Kobeissia, 1988; Antohe and Gladwell, 2004;
Celik Kizilkan and Aydin, 2006; Habib
and El-Zahar, 2008 and Gu et al.,2011).
In practical implementation of a numerical method for the solution of differential
equations the use of variable-length step-size is a crucial point because it
permits us to automatize the control of the error. Therefore, in this study,
we adapted MsDTM to define a new adaptive step-size semi-analytical and numerical
method. This new approach is applied directly to a nonlinear biochemical reaction
model, without requiring linearization or perturbation and the analytical-numerical
solutions of our model are obtained using piecewise convergent series with easily
computable components over a sequence of variable-length sub-intervals. Then,
a comparison between the present method, classical DTM, classical multi-step
DTM and the Runge-Kutta method is presented. The present method gives a more
accurate approximation with less computational effort than fixed step-size methods.
The results demonstrate reliability and efficiency of the method developed for
solving the considered problems.
BASIC CONCEPTS OF MULTI-STEP DIFFERENTIAL TRANSFORM METHOD
The MsDTM that has been developed for the analytical solution of the ODEs presented in this section for the systems of ODEs. For this purpose, we consider the following initial value problem for systems of ODEs:
subject to the initial conditions:
Let [t0, T] be the interval over which we want to find the solution
of the initial value problem Eq. (1-2).
In actual applications of the DTM, the Nth-order approximate solution of the
initial value problem (Eq. 1-2) can be expressed
by the finite series:
Equations 3 and 4 imply that the concept
of differential transformation is derived from the Taylor series expansion.
From the basic definition of the differential transformation, one can obtain
certain laws of transformational operations, some of these, are listed in Table
1. Using some fundamental operations of DTM, we have the following recurrence
where, Fi(k, X1, X2, ...., Xn) is the differential transform of the function fi(t, x1, x2, ...., xn) for i = 1, 2, ...., n.
The differential transform Xi(k) of the unknown functions xi(t) can be obtained by solving the iterating algebraic system (Eq. 5). In order to speed up the convergence rate and to improve the accuracy of resulting solutions, the entire domain [t0, T] is usually split into sub-intervals and the algorithm of MsDTM is applied as follows:
Assume that the interval [t0, T] is divided into M sub-intervals
[tm-1, tm], m = 1, 2, ...., M of equal step-size h = (T-t0)/M
by using the nodes tm = t0+mh. The main ideas of the MsDTM
are as follows:
First, we apply the DTM to the initial value problem (Eq. 1-2)
over the interval [t0, t1], we will obtain the following
using the initial conditions xi(t0) = ci.
|| Fundamental operations of differential transform method
For m≥2 and at each sub-interval [tm-1, tm], we will
use the initial conditions xi, m(tm-1) = xi, m-1(tm-1)
and apply the DTM to the initial value problem (Eq. 1-2)
over the interval [tm-1, tm]. The process is repeated
and generates a sequence of the approximate solutions xi, m(t), m
= 1,....., M, i = 1, 2, ..., n for the solutions xi(t):
Finally, the MsDTM assumes the following solution:
ADAPTIVE STEP-SIZE ERROR CONTROL ALGORITHM
Here, we present an adaptive step-size error control algorithm based on discussing some properties of the MsDTM.
MsDTM has several advantageous properties over the widely used classical methods.
One of the advantages of the MsDTM is that the approximate solution is given
as a piecewise polynomial function defined on the sub-intervals of the whole
interval and the local error of this solution, at the interior points of the
sub-interval, is less than that one at the end point. This property offers a
different facility for adaptive error control. Another advantage of the MsDTM
is that the truncated Taylor's series solution is obtained in explicit form
in every sub-interval. This property facilitates to investigate several properties
of the approximate solution such as the local error of approximate solution
for each component because the main term of the local error is known in the
form of |Xi, m(N)|τN where τ is the local time
variable in a sub-interval [tm, t]. The series term |Xi, m(N)|τN
can be viewed as a local error estimate for the series of degree N-1, which
is then applied to the more accurate series of degree N. Generally, the local
error should not be calculated from the next series term, |Xi, m(N+1)τN+1,
due to the extra computation required and the fact that it is not a reliable
error estimate (Corliss and Lowery, 1977).
So without any further calculation one can estimate the admissible step-size to ensure the prescribed local error by use just one coefficient of our obtained series solution. Using this technique while we apply MsDTM we get an automatic step-size control algorithm too. One simple step-size control algorithm could be the following.
||One gives the admissible local error δ>0 and chooses
the order N of the MsDTM
||From calculation, the values |Xi, m (N)|i = 1, 2,...,n, are
known for every solution component i
||At the grid point tm we calculate the value EN =
max(|Xi, m (N)|), i = 1, 2, ..., n
||We select such step-size hm for which hm = (δ/EN)1/(N)
and tm+1 = tm+hm
Another step-size selection may be just to use the last two coefficients of our series solution instead of only one, as mentioned above and imposing that both sets of coefficients are lower than the admissible local error that is:
The step selection methods above performed very similarly in the current study. Both methods provided stable, accurate solutions and used approximately the same number of time steps in head-to-head calculations.
We can conclude that for the present step-size control algorithm, there is no need for further calculations contrary to the Richardson extrapolation or embedded R-K algorithms. We have to remark that the step size control by Richardson extrapolation require 12 further function evaluation for a fourth order Rung-Kutta method. Also, there is no rejected step as occurs in any variable step formulation for Rung-Kutta or multistep methods because we choose the step-size once the series approximations are generated in order to obtain a required precision level.
Now, the present adaptive step-size error control algorithm is applied with MsDTM to obtain approximate analytical-numerical solutions of a nonlinear biochemical reaction system. We call this technique as adaptive step-size multi-step DTM (AsDTM).
APPLICATION TO NONLINEAR BIOCHEMICAL REACTION MODEL
Consider the well-known MichaelisMenten biochemical reaction model (Schnell
and Mendoza, 1997), i.e., the single enzyme substrate reaction scheme:
where, E is the enzyme, A the substrate, Y the intermediate complex and X the
product. The time evolution of scheme (10) can be determined from the solution
of the system of coupled nonlinear ODEs (Sen, 1988):
subject to the initial conditions:
where, the parameters k1, k-1 and k2 are positive
rate constants for each reaction. Systems (Eq. 10-12)
can be reduced to only two equations for A and Y. And in dimensionless form
of concentrations of substrate, x and intermediate complex between enzyme and
substrate, y, are given by Sen (1988):
subject to the initial conditions x(0) = 1 and y(0) = 0, where α, β
and ε are dimensionless parameters. For small value of ε, Eq.
13 becomes much stiffer or singularly perturbed and requires special numerical
treatments to solve it (El-Zahar and E-Kabeir, 2013).
By using the fundamental operations of DTM in Table 1, we
obtained the following recurrence relation to the Eq. 13:
The approximate solutions of the system (13) for ε = 0.1, α = 0.375,
β = 1.0 and tε[0.0, 0.4], using the AsDTM with N = 6 and δ =
0.001 are given as follows:
Figure 1 shows the obtained solutions of (13) for ε
= 0.1, α = 0.375, β = 1.0 over a narrow region, t ε[0, 0.4],
containing the boundary layer region, using DTM, MsDTM (N = 6, h = 0.1), AsDTM
(N = 6, δ = 0.001) and RK4 (h = 0.00001). We can see in Fig.
1 that the solutions obtained by DTM have a small interval of convergence
and deviates much from AsDTM solution and the numerical RK4 solution, while
the solutions obtained by AsDTM have wide intervals of convergence and agree
very well with RK4 solution.
||Comparison of the solutions of (a) substrate x and (b) substrate
y, obtained by the DTM, MsDTM, AsDTM and RK4 method
For the solutions obtained by MsDTM, the step size h = 0.1 is not sufficient
to approximate the solution over the boundary layer region where MsDTM needs
step size h<0.0453 to achieve the given admissible local error δ.
Figure 2 shows the obtained solutions of Eq. 13 using the AsDTM (N = 6, δ = 0.001) and the RK4 (h = 0.00001) for t ε[0, 5.0]. We can observe the high agreement between the solution obtained using the AsDTM and the one obtained using RK4 over a large interval.
The results obtained and shown in Fig. 1 and 2 confirm that the new approach, AsDTM, increases the interval of convergence for the series solution.
|| Approximate solutions of the Eq. 13 by
AsDTM and RK4 method
|| Time-step length used by AsDTM in solution of Eq.
Figure 3 shows the time-step length used for solving Eq. 13 by the AsDTM. We can observe that the given admissible local error δ = 0.001, is achieved by the AsDTM using 27 time-step, while the MsDTM needs step size h<0.0453 to achieve the given admissible local error δ and consequently needs at least 111 time-step.
The difference between the AsDTM solutions and the numerical RK4 solutions are given in Fig. 4 for different values of ε and δ where, Ey(t)|y(t)-yRK4| and Ex(t)|x(t)-xRK4|. We can observe that the solutions obtained using the AsDTM and the RK4 method are in high agreement for different values of ε and δ.
||Difference between the AsDTM solutions and numerical RK4 solutions
at different values of ε and δ
|| Comparison of processing time and time-step at different
values of δ and ε
Table 2 presents a comparison of processing time and time-step used in solving the system (13) by MsDTM and AsDTM to achieve a specific tolerance, δ, for different values of ε, where all calculations are carried out by MAPLE 14 software in a PC with a Pentium 2 GHz and 512 MB of RAM. We can observe that the AsDTM is a fast and effective tool for solving the considered problems using fewer time steps.
In this study, we adapted the multi-step DTM to define a new adaptive step-size semi-analytical and numerical method. This new approach provides analytical solutions in the form of piecewise convergent series with easily computable components over a sequence of variable-length sub-intervals. This approach is simple in applicability as it does not require linearization or perturbation like other numerical and approximate methods. We have applied it to the well-known Michaelis-Menten nonlinear biochemical reaction system and the obtained piecewise series solutions are presented. Numerical results are presented in figures and tables at different values of the tolerance δ and the perturbation parameter ε. The results confirm that the proposed method is a very accurate and efficient method compared with classical DTM, classical MsDTM, and RK4 method. The method works successfully in handling the biochemical reaction system with a minimum size of computations and a wide interval of convergence for the series solutions. This emphasizes the fact that the present method is applicable to many other nonlinear systems and it is reliable and promising when compared with the existing methods.
The author would like to thank the reviewers for their helpful comments which led to improvement of the manuscript.
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