Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 690587 | https://doi.org/10.1155/2012/690587

Zhaohua Gong, Chongyang Liu, Yongsheng Yu, "Modeling and Parameter Identification Involving 3-Hydroxypropionaldehyde Inhibitory Effects in Glycerol Continuous Fermentation", Mathematical Problems in Engineering, vol. 2012, Article ID 690587, 18 pages, 2012. https://doi.org/10.1155/2012/690587

Modeling and Parameter Identification Involving 3-Hydroxypropionaldehyde Inhibitory Effects in Glycerol Continuous Fermentation

Academic Editor: Gerhard-Wilhelm Weber
Received09 Jul 2012
Accepted21 Sep 2012
Published14 Nov 2012

Abstract

Mathematical modeling and parameter estimation are critical steps in the optimization of biotechnological processes. In the 1,3-propanediol (1,3-PD) production by glycerol fermentation process under anaerobic conditions, 3-hydroxypropionaldehyde (3-HPA) accumulation would arouse an irreversible cessation of the fermentation process. Considering 3-HPA inhibitions to cells growth and to activities of enzymes, we propose a novel mathematical model to describe glycerol continuous cultures. Some properties of the above model are discussed. On the basis of the concentrations of extracellular substances, a parameter identification model is established to determine the kinetic parameters in the presented system. Through the penalty function technique combined with an extension of the state space method, an improved genetic algorithm is then constructed to solve the parameter identification model. An illustrative numerical example shows the appropriateness of the proposed model and the validity of optimization algorithm. Since it is difficult to measure the concentrations of intracellular substances, a quantitative robustness analysis method is given to infer whether the model is plausible for the intracellular substances. Numerical results show that the proposed model is of good robustness.

1. Introduction

Microbial conversion of glycerol to 1,3-propanediol (1,3-PD) is particularly attractive in that the process is relatively easy and does not generate toxic byproducts. 1,3-PD has numerous applications in polymers, cosmetics, foods, lubricants, and medicines. Industrial 1,3-PD production has attracted attention as an important monomer to synthesize a new type of polyester, polytrimethylene terephthalate (PTT) [1]. However, compared with chemical routes, microbial production is difficult to obtain a high 1,3-PD concentration. The fermentation of glycerol by Klebsiella pneumoniae (K. pneumoniae) under anaerobic conditions is summarized in Figure 1. In the reductive pathway, 3-hydroxypropionaldehyde (3-HPA) is a toxic intermediary metabolite and its accumulation would arouse inhibitions to cells growth and to activities of the enzymes (such as glycerol dehydratase (GDHt) and 1,3-propanediol oxidoreductase (PDOR)) in glycerol metabolism [25].

It is critical to formulate the fermentation process using a precise mathematical model in the optimization of biotechnological processes. An excess kinetic model for substrate consumption and product formation was established in previous studies [710]. The models have been studied for parameter identification [11] and optimal control [1214] in fed-batch fermentation process. However, the intermediate and intracellular substances or enzymes of glycerol metabolism are not taken into consideration in those models. In fact, some important intermediate substances (such as 3-HPA), intracellular substances (such as 1,3-PD), and enzymes GDHt and PDOR play significant roles in glycerol metabolism. A mathematical model of glycerol fermentation concerning enzyme-catalytic reductive pathway and transports of glycerol and 1,3-PD across cell membrane was established in [6]. Although the achieved results are interesting, the effect of 3-HPA on cells growth is ignored. Moreover, that model is based on an assumption that 3-HPA inhibits the activities of the enzymes GDHt and PDOR all the time. In fact, there exists inhibitory effect of 3-HPA on cells growth owing to its toxicity. In addition, only when the accumulation of 3-HPA reaches some critical concentration, the inhibitions to enzymes can occur [3, 4].

Robustness is one of the fundamental characteristics of biological systems. By saying that a system is robust we imply that a particular function or characteristic of the system is preserved despite changes in the operating environment [15]. For robust biological systems, we expect that mathematical models attempting to explain these systems should also be robust [16]. In this paper, we are interested in the robustness to variations in kinetic parameters and use it to validate the plausibility of the mathematical model. This topic has been studied by the sensitivity analysis technique [1719], that is, repeated simulations by varying one parameter while holding all others fixed. However, single parameter insensitivity may not be sufficient owing to interactions between several parameters. Therefore, new methods are needed for studying multiparameter robustness.

Considering 3-HPA inhibitions to cells growth and to activities of the enzymes GDHt and PDOR in glycerol metabolism, we propose a novel mathematical model to describe 1,3-PD production by K. pneumoniae in continuous cultures. Some properties of the model, such as existence and uniqueness of the solution, continuity of the solution in kinetic parameters, and compactness of the set of feasible parameters, are discussed. Furthermore, a parameter identification model is established to determine the kinetic parameters in the presented system. Basing on the penalty function technique and an extension of the state space method, an improved genetic algorithm (GA) is then constructed to solve the identification model. Numerical example shows the appropriateness of the proposed system and the validity of optimization algorithm. Finally, a quantitative robustness analysis method is given to infer whether the model is robust, and numerical result shows that the proposed system is of good robustness.

This paper is organized as follows. In Section 2, the kinetic model is formulated to describe continuous fermentation process, whose important properties are also discussed. In Section 3, a parameter identification model is presented and an optimization algorithm is developed. Section 4 explores the robustness analysis of the proposed dynamical system. Finally, conclusions are provided.

2. Mathematical Model in Continuous Culture and Its Properties

2.1. Mathematical Model

During continuous fermentation of glycerol metabolism by K. pneumoniae under anaerobic conditions, glycerol is fed to the reactor continuously. As stated in [6], we assume that(H1) the transport of extracellular glycerol across cells membrane by passive diffusion and by glycerol transport facilitator;(H2) intracellular 1,3-PD is expected to be diffused from the intracellular environment to the extracellular medium in the fermentative broth.

Let be the state vector, and , , respectively denote the concentrations of biomass, extracellular glycerol, extracellular 1,3-PD, extracellular acetic, extracellular ethanol, intracellular glycerol, intracellular 3-HPA, and intracellular 1,3-PD at time in reactor. is the dilution rate, and is the initial glycerol concentration in feed.

Since 3-HPA is a toxic intermediary metabolite, its inhibition to the specific cellular growth rate is introduced besides substrate and products inhibitions. Therefore, the specific cellular growth rate [6] is modified as Under anaerobic conditions at 37°C and pH 7.0, the maximum specific growth rate, , and Monod constant, , are 0.67 and 0.28 mmol  , respectively. The critical concentrations , , are 2039, 1036, 1026, 360.9, and 300 mmol [6], respectively.

The specific consumption rate of substrate , and the specific formation rates of products and are expressed by the following equations based on previous works [6, 8, 10]

The governing equations based on mass balance [6, 10] can still be used to describe the concentrations of substrate and products for glycerol metabolism, for example, biomass, extracellular glycerol, extracellular 1,3-PD, acetate, ethanol, and intracellular glycerol which are described by (2.3)–(2.7), respectively

The reductive pathway is emphasized because 3-HPA is the key intermediate for 1,3-PD production. 3-HPA accumulation during fermentation process can cause growth cessation and low product formation [2, 4]. Moreover, when the accumulation of 3-HPA reaches some critical concentration, the inhibitions to enzymes GDHt and PDOR can occur [3, 4]. So the intracellular concentration change of 3-HPA can be described by where , are 0.53 and 0.14 mmol  [20, 21], respectively. is the critical concentration of 3-HPA beyond which the inhibitions to the activities of GDHt and PDOR occur. Moreover, in (2.8), and are the specific activities of GDHt and PDOR in vitro, which can be described by the following equations:

The intracellular 1,3-PD concentration depends on the conversion of 3-HPA catalyzed by PDOR whose activity is inhibited by the substrate, the diffusion from the intercellular to the extracellular and the dilution effect on cell growth, so whose variation can be formulated by

Now, let be the kinetic parameter vector to be identified. Denote and let the right-hand sides of (2.3)–(2.7), (2.8), and (2.11) be . Then, the proposed mathematical model can be rewritten as the following nonlinear dynamical system: where is the steady-state moment of the continuous fermentation process.

2.2. Properties of the Dynamical System

To begin with, we introduce some symbols which will be used below. Let be the admissible set of the kinetic parameter vector . Let and , denote the lower and upper bounds of the state vector , respectively. Let be the admissible set of , and let be the admissible set of initial glycerol concentration in feed medium and dilution rate .

For the system (2.12), we assume that(H3) the set is a nonempty bounded closed set;(H4) the absolute difference between extracellular and intracellular 1,3-PD and that of glycerol concentration is bounded, that is, and such that

Under the assumptions (H3) and (H4), we can easily verify the following properties of the velocity vector field .

Property 1. For any and , the function is locally Lipschitz continuous in on .

Property 2. For any and , the function satisfies linear growth condition, that is, there exist constants such that where is Euclidean norm.

Proof. For given and , let , , , , , , , . Let , , , , . Then, we can obtain Finally, set , then we have (2.14) holds with . The proof is completed.

Then, the existence and uniqueness of the solution for the system (2.12) can be confirmed in the following theorem.

Theorem 2.1. For any and , the system (2.12) with given initial state has a unique solution denoted by . Moreover, is continuous in on .

Proof. The proof can be obtained from Properties 1, and 2 and the theory of ordinary differential equations [22].

Given , we define the solution set of the system (2.12) as follows: Since the concentrations of biomass, glycerol, 3-HPA, and products are restricted in during the actual continuous cultures, the set of admissible solutions is Furthermore, let the set of feasible parameter vectors corresponding to be

From Theorem 2.1 and the above definitions, we have the following result.

Theorem 2.2. The feasible parameter set defined by (2.19) is a compact set.

Proof. In view of the compactness of , we obtain that is a bounded set. Moreover, for any sequence , there exists at least a subsequence such that as . It follows from Theorem 2.1 that and . Since is a compact set, we must conclude that , which implies the closeness of set . The proof is completed.

3. Parameter Identification Model and Optimization Algorithm

3.1. Parameter Identification Model

Now, we determine the kinetic parameter in by constructing an identification problem as follows.

Let be the total number of experiments carried out under different dilution rates and initial glycerol concentrations. For given , , we have the experimental steady-state data of extracellular substances in continuous cultures. Denote the steady-state concentrations of biomass, extracellular glycerol, extracellular 1,3-PD, acetate, and ethanol as , correspondingly. Let .

In particular, the stable state of a dynamical system (2.12) in the following identification problem is actually referred to the approximate stability defined as follows.

Definition 3.1. For given , a state vector is said to be an approximately stable solution within a precision if there exists such that is a solution of the system (2.12) satisfying where is the Euclidean norm, and .

Since the orders of magnitude for concentrations involved are different, we adopt the average relative error between the computational values and the experimental data at steady-state moments , as the criterion

To determine the parameter values of the system (2.12), a parameter identification problem, in which is taken as the cost function, can be formulated as

Following the above properties, we can conclude the following theorem.

Theorem 3.2. For , , there exists in such that .

3.2. Optimization Algorithm

In , the constraint actually involves the constraint of continuous state, that is, . In this section, we will develop a computational method for solving our proposed parameter identification problem . By means of the penalty function technique and an extension of state space method, we transcribe into an optimization problem only with box constraint. First of all, we introduce a new state variable satisfying where , and .

Obviously, the process satisfies the constraint of continuous state if and only if . Furthermore, denote the cost function in by where and are penalty factors. Then can be rewritten as

As a result, is an optimization problem only with the box constraint and equivalent to as and .

Since is nondifferentiable, we construct an improved genetic algorithm (GA) to solve taking advantage of the problem’s characteristic. Let . In the improved genetic algorithm, we take as the individual, and (3.4) as the fitness function. Now, we describe the algorithm in detail as follows.

Algorithm 3.3. We have the following steps: Step 1. Initialize population size , the maximal iterations , penalty factors and , precision , parameters , and . Set and randomly generate individuals by uniform distribution, that is, . Let .Step 2. Compute the fitness value by taking into the systems (2.12) and (3.3).Step 3. Generate the crossover offspring by arithmetic crossover and evaluate their fitness values .Step 4. The individual produced by crossover is operated with normality variation until the produced one is in . The mutation offspring is denoted by . Compute their fitness values .Step 5. Form the next generation by selecting the best individuals from the ones.Step 6. Set , and , if either or no progress is made in the last generations, then output the best individual and stop, otherwise go to Step 3.

3.3. Numerical Results

According to the actual continuous fermentation process, the initial state g ,  mmol and 22 groups of the experimental steady-state data are used. Here, 10 groups of experimental steady-state data under substrate-limited conditions and 12 groups of experimental steady-state data under substrate-sufficient conditions. The critical value is taken as the value in [23]. Moreover, the admissible set of parameter vectors is taken as the decrements and increments of 0.9 times the kinetic parameter values in [6]. By applying Algorithm 3.3, we obtained the optimal parameter vector of the system (2.12) under the substrate-limited and the substrate-sufficient conditions shown in Table 1. Accordingly, the cost function values, respectively, are and under the above two cases. Here, all the computations are performed in Visual C++ 6.0 and numerical results are plotted by MATLAB 7.10.0. In particular, the ODEs in the computation process are numerically calculated by improved Euler method [24] with the relative error tolerance . The parameters used in Algorithm 3.3 are , , , , , and , respectively. It should be noted that these parameters are derived empirically after numerous experiments. The comparison of three extracellular substances concentrations, that is, biomass, extracellular glycerol, and extracellular 1,3-PD, between experimental steady-state data and computational results under substrate-limited conditions are shown in Figures 2, 3, and 4. Furthermore, The comparisons of three extracellular substance concentrations, that is, biomass, extracellular glycerol, and extracellular 1,3-PD, between experimental steady-state data and computational results under substrate-sufficient conditions are also shown in Figures 5, 6, and 7. From the above figures, we can see that the simulation results can approximate the experimental steady-state data well. Thus, the mathematical model considering 3-HPA inhibitions to cells growth and to activities of enzymes can well describe the continuous fermentation process.


Conditions Optimal parameter vector

(16.93, 0.015, 375.9, 0.134, 9.5, 162.8,
0.013, 0.014, 0.518, 0.135, 6.239, 203.6,
Substrate-limited condition 37.5, 0.042, 18.43, 3.13, 0.006489,18.83,
12.82, 1.843, 52.03, 10.91, 5.045, 2.983,
0.567, 0.087, 1.852, 6.225, 0.346, 0.138)

(36.37, 0.287, 456, 0.878, 0.5, 162.8,
  0.036, 0.087, 2.235, 2.216, 11.25, 282.1,
Substrate-sufficient condition  39.12, 0.629, 9.841, 3.854, 0.006678,
  13.09, 10.77, 0.590, 62.83, 7.833, 17.39,
  5.624, 0.353, 0.6145, 0.836, 8.094, 1.254, 0.138)

4. Robustness Analysis of the Model

For robust biological systems, we expect that mathematical models that attempt to explain these systems should also be robust. Robustness of the model is analyzed in this section. In view of the glycerol dissimilation mechanism, we assume that(H5) for each , and given , there exists an approximate stable solution of the system (2.12).

4.1. Mathematical Measurement of Robustness

Robustness can be defined as a system’s characteristic that maintains one or more of its functions under external and internal perturbations [15]. In this study, the substance concentrations at steady-state moments are viewed as the quantitative descriptors of the system and the perturbations are the parameters variations in .

Let be a finite set whose elements are drawn from by random distribution. Since the state vector of the system (2.12) takes as the input parameter, we define the representative steady-state vector as where and .

Let be the feasible space of the perturbation parameter vector, where is sufficiently small positive number to measure the magnitude of the parameter disturbances. Furthermore, let be a perturbation set which is composed of elements randomly generated from . Denote the set of the representative steady-state vector corresponding to the varied parameters by For the representative steady-state vectors in , we are interested in their deviations from the representative steady-state vector corresponding to the optimal parameter vector . Define the expectation of the th component of these deviations as follows:

Based on the above analyses, a mathematical definition of biological robustness can be stated as follows.

Definition 4.1. The robustness measurement of a model with regard to steady states against a set of parameter perturbations is where is the state index set involved in robustness analysis.

4.2. Algorithm and Numerical Results

On the basis of the definition of robustness, we develop an Algorithm 4.2 to compute the robustness to variations of the optimal kinetic parameter vector .

Algorithm 4.2. We have the following steps:Step 1. Generate perturbation parameter vectors by random distributions domain.Step 2. Compute the representative steady-state vectors corresponding to optimal parameters and to perturbation parameters and according to (4.1).Step 3. Evaluate the robustness by the mathematical measurement defined in (4.5).Step 4. Repeat Step 1, Step 2, and Step 3 for times and compute the expectation value of the generated sequence .

According to Algorithm 4.2, we have investigated the robustness of the system (2.12) for the intracellular substances. Here, the number of perturbation parameters , precision , the set , and the repeating times take values 200, 0.05, , and 100, respectively. Computational results for the substrate-limited and substrate-sufficient cases are listed in Table 2. Furthermore, the simulation curves for intracellular glycerol, 3-HPA and intracellular 1,3-PD under the substrate-limited and the substrate-sufficient conditions are illustrated in Figures 8, 9, 10, 11, 12, and 13. From Table 2 and the numerical results, we can see that the proposed model is of good robustness.


Conditions Substrate-limitedSubstrate-sufficient

Values 3.972682.70685

5. Conclusions

Glycerol bioconversion to 1,3-PD by K. pneumoniae in continuous cultures under anaerobic conditions was investigated. Contrasting with the existing models, the paper proposed a new mathematical model by considering 3-HPA inhibitory effects on cells growth and on the activities of the enzymes GDHt and PDOR. Then, we discussed some properties of the system. Furthermore, we presented a parameter identification model to determine the kinetic parameters in the presented system. Since the identification model is subject to constraint of continuous state, we developed a computational approach to solve the parameter identification model based on the penalty function technique and an extension of the state space method. More importantly, a quantitative robustness analysis method was given to infer whether the model is plausible. Numerical results showed that the proposed system is of good robustness.

Acknowledgments

The supports of the Natural Science Foundation for the Youth of China (nos. 11001153, 11201267), the Tianyuan Special Funds of the National Natural Science Foundation of China (no. 11126077), the Shandong Province Natural Science Foundation of China (nos. ZR2010AQ016, ZR2011AL003, and BS2012DX025) and the Fundamental Research Funds for the Central Universities (no. DUT12LK27) are gratefully acknowledged.

References

  1. H. Biebl, K. Menzel, and A.-P. Zeng, “Microbial production of 1,3-propanediol,” Applied Microbiology and Biotechnology, vol. 52, no. 3, pp. 289–297, 1999. View at: Google Scholar
  2. F. Barbirato, J.-P. Grivet, P. Soucaille, and A. Bories, “3-Hydroxypropionaldehyde, an inhibitory metabolite of glycerol fermentation to 1, 3-propanediol by enterobacterial species,” Applied and Environmental Microbiology, vol. 62, no. 4, pp. 1448–1451, 1996. View at: Google Scholar
  3. J. Hao, R.-H. Lin, Z.-M. Zheng, Y. Sun, and D.-H. Liu, “3-Hydroxypropionaldehyde guided glycerol feeding strategy in aerobic 1,3-propanediol production by Klebsiella pneumoniae,” Journal of Industrial Microbiology and Biotechnology, vol. 35, no. 12, pp. 1615–1624, 2008. View at: Google Scholar
  4. F. Barbirato, E.-H. Himmi, T. Conte, and A. Bories, “1,3-propanediol production by fermentation: an interesting way to valorize glycerin from the ester and ethanol industries,” Industrial Crops and Products, vol. 7, no. 2-3, pp. 281–289, 1998. View at: Google Scholar
  5. W. Wang, J. B. Sun, M. Hartlep, W.-D. Deckwer, and A.-P. Zeng, “Combined use of proteomic analysis and enzyme activity assays for metabolic pathway analysis of glycerol fermentation by Klebsiella pneumoniae,” Biotechnology and Bioengineering, vol. 83, no. 5, pp. 525–536, 2003. View at: Google Scholar
  6. Y.-Q. Sun, W.-T. Qi, H. Teng, Z.-L. Xiu, and A.-P. Zeng, “Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: concerning enzyme catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane,” Biochemical Engineering Journal, vol. 38, pp. 22–32, 2008. View at: Google Scholar
  7. A.-P. Zeng, A. Rose, H. Biebl, C. Tag, B. Guenzel, and W. D. Deckwer, “Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation,” Biotechnology and Bioengineering, vol. 44, no. 8, pp. 902–911, 1994. View at: Google Scholar
  8. A.-P. Zeng and W.-D. Deckwer, “A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions,” Biotechnology Progress, vol. 11, no. 1, pp. 71–79, 1995. View at: Google Scholar
  9. A.-P. Zeng, “A kinetic model for product formation of microbial and mammalian cells,” Biotechnology and Bioengineering, vol. 46, no. 4, pp. 314–324, 1995. View at: Google Scholar
  10. Z.-L. Xiu, A.-P. Zeng, and L.-J. An, “Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol,” Journal of Dalian University and Technology, vol. 40, pp. 428–433, 2000. View at: Google Scholar
  11. Z.-H. Gong, “A multistage system of microbial fed-batch fermentation and its parameter identification,” Mathematics and Computers in Simulation, vol. 80, no. 9, pp. 1903–1910, 2010. View at: Publisher Site | Google Scholar
  12. Z.-H. Gong, C.-Y. Liu, E.-M. Feng, L. Wang, and Y.-S. Yu, “Modelling and optimization for a switched system in microbial fed-batch culture,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3276–3284, 2011. View at: Publisher Site | Google Scholar
  13. C.-Y. Liu, Z.-H. Gong, and E.-M. Feng, “Modeling and optimal control of a nonlinear dynamical system in microbial fed-batch fermentation,” Mathematical and Computer Modelling, vol. 53, no. 1-2, pp. 168–178, 2011. View at: Publisher Site | Google Scholar
  14. C.-Y. Liu, Z.-H. Gong, E.-M. Feng, and H.-C. Yin, “Optimal switching control of a fed-batch fermentation process,” Journal of Global Optimization, vol. 52, no. 2, pp. 265–280, 2012. View at: Publisher Site | Google Scholar
  15. H. Kitano, “Towards a theory of biological robustness,” Molecular Systems Biology, vol. 3, p. 137, 2007. View at: Google Scholar
  16. T.-H. Tian, “Robustness of mathematical models for biological systems,” The ANZIAM Journal, vol. 45, pp. C565–C577, 2004. View at: Google Scholar
  17. M. Morohashi, A.-E. Winn, M.-T. Borisuk, H. Bolouri, J. Doyle, and H. Kitano, “Robustness as a measure of plausibility in models of biochemical networks,” Journal of Theoretical Biology, vol. 216, no. 1, pp. 19–30, 2002. View at: Publisher Site | Google Scholar
  18. R.-S. Erb and G.-S. Michaels, “Sensitivity of biological models to errors in parameter estimates,” Pacific Symposium on Biocomputing, vol. 4, pp. 53–64, 1999. View at: Google Scholar
  19. M.-A. Savageau, “The behavior of intact biochemical control systems,” Current Topics in Cellular Regulation, vol. 6, pp. 63–130, 1972. View at: Google Scholar
  20. S. Honda, T. Toraya, and S. Fukui, “In situ reactivation of glycerol inactivated coenzyme-B12-dependent glycerol dehydratase and dioldehydratase,” Journal of Bacteriology, vol. 143, pp. 1458–1465, 1980. View at: Google Scholar
  21. R. Daniel, R. Boenigk, and G. Gottschalk, “Purification of 1,3-propanediol dehydrogenase from Citrobacter freundii and cloning, sequencing, and overexpression of the corresponding gene in Escherichia coli,” Journal of Bacteriology, vol. 177, pp. 2151–2156, 1995. View at: Google Scholar
  22. D.-K. Arrowsmith and C.-M. Place, Ordinary Differential Equations, Chapman & Hall, London, UK, 1982.
  23. B.-Y. Shen, C.-Y. Liu, J.-X. Ye, E.-M. Feng, and Z.-L. Xiu, “Parameter identification and optimization algorithm in microbial continuous culture,” Applied Mathematical Modelling, vol. 36, no. 2, pp. 585–595, 2012. View at: Publisher Site | Google Scholar
  24. E. Hairer, S.-P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I, vol. 8, Springer, Berlin, Germany, 1993.

Copyright © 2012 Zhaohua Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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