1. Introduction

Since distributed fiber Raman amplifiers (DFRAs) have been suggested as an enabling technology for long-haul high-capacity wavelength-division multiplexing (WDM) systems [1–4], the optimizations for the signal gain profile and noise performance have attracted extensive interest [5–10]. Unfortunately, longer wavelength channels have larger output OSNR than shorter wavelength channels in the case of pure backward-pumped Raman amplifiers [11, 12]. But co-propagating Raman pumping can offer lower equivalent noise figures [13], and reduce multipath interference noise induced by signal double-Rayleigh scattering when combined with counter-propagating pumping [3, 13]. The noise performance can be effectively improved by the additional forward-pump source. Therefore, more and more transmission systems with bi-directionally pumping DFRAs have been investigated and reported by the simulations and experiments [5, 6].

Although the optimizations for the gain spectrum and noise performance have been investigated widely, most of them are independently implemented (i.e., the gain spectrum or noise performance is optimized separately). For instance, only the Raman gain is optimized in Refs.[10, 11, 14, 15], and the noise and nonlinear performance are investigated in Refs.[7, 9]. In most reported results, moreover, the signal gain spectrum is optimized only for pure backward pumping schemes instead of bi-directionally pumping schemes by using all kinds of algorithms (e.g., the neural network method [14], simulated annealing algorithm [15] and genetic algorithm (GA) [16]). The key reasons are that there exists somewhat difficulty in simulating bi-directional multi-pump DFRAs with higher pump power and longer fiber span [17].

Kado and Emori et al simultaneously realized the flatness of Raman gain and noise figure in a bi-directionally pumped Raman amplifier [6, 18], but there are eight pump diodes (five backward- and three forward- pump diodes) for the gain flatness of 0.5 dB and noise flatness of 0.7 dB at the on-off gain of 9 dB. And three additional pump diodes were employed in order to decrease the flatness of noise figure from 2.6 dB to 0.7 dB [6]. So, it is far from the optimization. In this paper, a novel optimizing algorithm based on the geometry compensation technique, multiple shooting algorithms and hybrid-GA is proposed for optimizing the profile of Raman gain and OSNR. To increase computing efficiency, a fast numerical method is proposed also. Based on the proposed methods, the DFRAs with co-, counter- and bi-directionally pumping cases are optimized and compared in detail. The simulations demonstrate that bi-directionally pumping schemes not only can increase the gain spectrum performance but also can equalize OSNR tilt.

2. Mathematical model for distributed fiber Raman amplifiers

In calculating the DFRA gain profile, such noise effects as spontaneous Raman scattering, Rayleigh backscattering and thermal factor, etc., can be reasonably skimmed, and then the major influence is the interactions of pump-to-pump, signal-to-signal and pump-to-signal, as well as the attenuation [10, 19]. In the steady state, the coupled nonlinear equations for simulating Raman gain spectrum can be described as [10, 11, 15, 19]

where, the indexes k=1, 2,…,l and k=l+1,…,l+m (l=5 and m=45 in this article) represent pump and signal waves, respectively, and the frequencies are numerated in decreasing order.

However, amplified spontaneous emission (ASE) and Rayleigh backscattering effect are two main aspects of DFRA noise performance, and their impairments impose limits on the maximum allowable distributed gain in a system [20]. Since backscattering powers of pumps and signals are less by about 30 dB and 20 dB than their original powers, respectively [27], the backscattering pumps and backscattering signals can be reasonably ignored in simulating noise waves. Then the model equations for noise waves include such physical effects as attenuation, stimulated Raman scattering, spontaneous Raman scattering, Rayleigh scattering, thermal noise, and so on, which can be expressed in terms of the following equations [9, 21, 22, 29]

Here, P_k , v_k , η_k and α_k are the pump (or signal) power, frequency, Rayleigh scattering and attenuation coefficient for the kth wave, respectively. P _n,k is the noise power in the bandwidth Δv. P_j =$P_j^{\pm }$ +$P_j^{\pm }$ . The superscript ‘+’ and ‘-’ denote forward- and backward-propagating pump, signal and noise waves, respectively h, k_B , T and A_eff are Planck’s constant, Boltzmann constant, temperature and effective area of optical fiber, respectively. The factor of Γ accounts for polarization randomization effects. g_R (v_j -v_k ) is the Raman gain coefficient from the frequency v_j to v_k .

3. Novel algorithms

Because the strong Raman interactions of pump-to-pump, signal-to-signal and pump-to-signal make the optimal design of DFRAs with required gain spectra become somewhat difficult, the DFRA design presents a grand challenge to the optimization algorithm. So, the neural network method, simulated annealing algorithm and GA are utilized to optimize the design of DFRAs [14–16]. Although GA can offer a powerful tool for optimizing the design of DFRAs, its convergence is somewhat slow due to its intrinsic shortcoming. However, the geometry compensation technique can greatly effectively increase the convergence speed of optimizing process [23]. But this technique only provides an approximate solution for DFRA optimization and is lack of capability searching the global solution.

On the other hand, the design of multipump Raman amplifiers involves the multimodal function optimization problems [16], so that the traditional algorithms fail to optimize their designs if expecting multiple global or near-global solutions. But the hybrid-GA can overcome such difficulty [16]. When taking into account the noise performance, different global or near-global solution has different OSNR. To simultaneously realize the flatness of both Raman gain profile and OSNR spectra, a novel algorithm with fast convergence speed is proposed as follow.

Step 1: Based on the geometry compensation technique in our previous reports [23], the approximate optimization solution for the gain profile can be rapidly calculated from Eq. (1).

Step 2: Under the obtained parameters of the solution in Step 1, the multiple global or near-global solutions can be found from the hybrid-GA when only considering the gain spectrum [16].

Step 3: Based on the multiple shooting algorithms [17] and the parameter set of the obtained gain profile in Step 2, the noise waves including Raman scattering, Rayleigh scattering, thermal noise, etc., can be calculated from Eq. (2). Taking into account both the gain profile and noise performance, the optimized global solution with the flatness of both Raman gain and OSNR can be obtained.

The simulated results exhibit that ① the novel algorithm can improve the optimizing efficiency very much in comparison with pure hybrid-GA [16, 28], and ② the difficulty in solving the coupled equations of bi-directional DFRAs with longer fiber distance, higher pump power and multi-pump diodes is completely overcome.

4. Fast numerical method

It is obvious that Eqs. (1) and (2) are the two-point boundary value problems, and they are usually solved by the shooting algorithms. Because the “pure” shooting algorithm has faster computing speed and the multiple shooting algorithms have more stability [17], they are used in our simulations. In the DFRA design, however, it will usually need exhaustive computing time to obtain well-behaved results if using a direct integration of its coupled nonlinear equations. Then, the simulation method has to provide a reasonable computational time for optimizing the design of DFRA. To increase the computing speed, here, a fast numerical algorithm with automatic step adjustment is proposed for Eq. (1), i.e.,

The function F in Eqs. (3) is from Eq. (1.b) and h denotes the step size (h=z _j+1-z_j ). The error estimate E can be calculated from the norm of the difference between Eqs. (3.h) and (3.i), i.e.,

To compare the proposed numerical method with traditional Dormand-Prince method, Appendix-1 shows their differences in detail. Because the detailed deduction process of Eqs. (3) and (4) is beyond the research range of this paper, it is not covered in detail here. However, their source codes are given in Appendix-2.

5. Optimizations and results

In the following simulations, we assume that A_eff =80 µm², Γ=2, η=7×10^-8 m^-1 and the span distance L=80 km; there are 45 signal channels spaced 200 GHz from 1539.5 to 1612.5 nm (the corresponding bandwidth Δλ=73 nm); the signal power of each channel is -10 dBm; the gain spectrum g_R (Δv) and attenuation spectrum α(v) of the fiber are shown in Fig. 1; the gross Raman gain can compensate the loss of signals (i.e., the net gain is more than 0 dB). Five pump diodes are fixed and six type structures are optimized.

 

Fig. 1. Raman gain spectrum g_R (Δv) of the fiber, and its attenuation spectrum α(v) in the window of 1380–1700 nm.

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The optimized results for pump and signal power evolutions along the fiber distance are demonstrated in Fig. 2, where (a), (b), (c), (d), (e) and (f) are the configurations of 5 backward pump diodes (Type-1), 1 forward and 4 backward pump diodes (Type-2), 2 forward and 3 backward pump diodes (Type-3), 3 forward and 2 backward pump diodes (Type-4), 4 forward and 1 backward pump diodes (Type-5), and 5 forward pump diodes (Type-6), respectively. Conveniently, above six configurations are named as Type-1, Type-2, Type-3, Type-4, Type-5 and Type-6, respectively. Pj (j=1, 2, …, 5) in the legends of Fig. 2 are the optimized pumps, and their corresponding pumps and wavelengths are tabulated in Table 1. From Table 1, one can see that Type-1 (i.e., pure backward pumping scheme) has the least total power of input pumps (i.e., 771.6 mW), while Type-4 has the greatest power (i.e., 875 mW) in six structures.

Table 1. Optimized parameter set for Fig. 2

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Fig. 2 exhibits that ① Type-2 has the least fluctuation, which leads to the least ripple of OSNR (see Fig. 3), for all signals along the transmission distance; ② Type-4 and Type-1 have the least and greatest power ripples for the output signals, respectively. Their detail signal spectra at the output port (i.e., z=80 km) are exhibited in Fig. 3, where the triangular symbols represent the signal channels. Based on the optimized pump and signal values along the transmission fiber, the corresponding OSNRs are calculated from Eq. (2), and the red curves in Fig. 3 are the fitted lines for OSNRs.

 

Fig. 2. Pump and signal power evolutions along the fiber distance, where (a), (b), (c), (d), (e) and (f) are the configurations of Type-1, Type-2, Type-3, Type-4, Type-5 and Type-6, respectively.

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Table 1 and Fig. 3 illustrate that the net gain ripples of signals with the bandwidth of 73 nm are 1.02, 0.76, 0.80, 0.74, 0.86 and 1.04 dB, OSNR ripples are 2.37, 0.81, 2.17, 2.22, 1.26 and 1.17 dB, and their corresponding slopes of fitted lines are 0.0324, 0.0014, -0.031, -0.0342, -0.0175 and 0.0044 dB/nm, respectively, for Type-1, 2, 3, 4, 5 and 6. Different pumping structure (i.e., Type-j, j=1,…,6) has different power profile along the transmission fiber see Fig. 2), which causes different gain ripple (see Table 1 and Fig. 3).

 

Fig. 3. Relationships of OSNR and output signal power with the wavelength, where (a), (b), (c), (d), (e) and (f) for Type-1, 2, 3,4, 5 and 6, respectively. The red curves are the fitted lines of OSNR.

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From Figs. 2 and 3, we can find that ① pure forward pumping scheme (i.e., Type-6) can offer more than 3-dB OSNR advantage by comparing pure backward pumping scheme (i.e., Type-1); ② bi-directionally pumping cases (i.e., Type-2, 3, 4 and 5) have less ripples of net gain and OSNR than pure backward pumping case; ③ bi-directionally pumping configuration with appropriate forward power can not only increase the performances of gain and OSNR but also equalize the OSNR tilt in comparison with pure backward pumping configuration, e.g., the slope of fitted line of OSNR is 0.0014 dB/nm for Type-2 instead of 0.0324 dB/nm for Type-1.

In the real optimization of DFRAs, besides OSNR, nonlinear impact should be taken into account. They can be estimated by overall nonlinear phase-shift along the span length [5, 7, 9], which is defined as K_NL =γ·${\int }_0^L$ P_s (z)dz, where γ is the nonlinear coefficient (γ=1.3 W^-1km^-1 in this paper [24]) and P_s (z) is the signal power at position z. Based on the optimized signal profile in Fig. 2, K_NL is calculated and shown in Fig. 4. One can see, from Fig. 4, that ① pure backward pumping (i.e., Type-1) has the least K_NL , whereas pure forward pumping (i.e., Type-6) has the largest; ② K_NL of Type-2 is close to that of Type-1; ③ maximum K_NL of Type-6 is greater by a factor of >10 than that of Type-1; ④ increasing amount of forward pumping power leads to greater K_NL (e.g., K_NL of Type-5 is more than that of Type-4). Take into account K_NL and relative intensity noise (RIN) [25, 26], therefore, more forward pumping power is not encouraged in the design of DFRAs.

 

Fig. 4. Relationship between K_NL and signal wavelength.

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Fig. 5. Profiles of signal power and OSNR for global or near-global optimum in Type-2.

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By comparing six different pumping structures in Figs. 2–4, we can see that ① Type-6 has the greatest impact of the Kerr nonlinearity although it can offer more than 3-dB improvement of OSNR in comparison with Type-1; ② although Type-4 can provide the least ripple of net gain, its OSNR ripple and K_NL are much greater than Type-2 and Type-4 costs more input pump power (i.e., 875 mW) than any other structure. Taking into account the system cost and complexity, the tradeoff between noise and nonlinear performance of DFRAs has to be implemented. Therefore, Type-2 is suggested to the best candidate for six pumping schemes, which can offer not only less ripples of both gain and OSNR but also lower nonlinearity.

Because the design of multi-pump Raman amplifiers is the multimodal function optimization problem and hybrid-GA can offer multiple global or near-global maximums when only taking into account signal gain performance [16], we can designate one of maximums with the least noise as the best solution. In Type-2, for instance, we obtain 6 global or near-global solutions for the signal gain spectrum, whereas these solutions have different OSNR ripples. They are plotted in Fig. 5 and are tabulated in Table 2 (the parameters of the best solution are illustrated in Table 1 and N_i (i=1, 2, …, 5) represents the i-th global or near-global solution). From Table 2 and Figs. 2–5, one can see that, although all maximums have the approximately same gain ripple (i.e., 0.76 dB or 0.77 dB), the solution with the least OSNR ripple (i.e., 0.81 dB) are demonstrated in Fig. 3. Fig. 5 shows that different global solution has different profiles of gain and OSNR although their gain ripples are the approximate same.

Table 2. Optimized parameter set for Fig. 5

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6. Discussions

6.1. Comparing to the reported results

Compared to the simulation and experiment results in [6, 18], our results can obviously offer better solutions for designing DFRAs. For instance, not only can the number of laser diodes be reduced from 8 to 5, but also the ripples of both gain and noise performance are decreased (Type-2 significantly decreases the ripples of both gain and OSNR, whereas only OSNR ripple is decreased under the approximate same gain ripple in Refs.[6, 18]).

Although the noise performance of DFRAs has been investigated widely, they are usually deal with by the independent noise terms (e.g., ASE and double Rayleigh backscattering are calculated separately) [5, 7, 9]. Furthermore, the double Rayleigh backscattering of the signals is simulated under the assumption of undepleted pump approximation. Obviously, such approximation is not consistent with the real situation, which can clear be seen from Fig. 2. Then, the obtained noise is not accurate enough under the undepleted pump approximation. In this paper, based on the multiple shooting algorithms, the noise is calculated directly from the rigorous Raman amplifier propagation equations and the accuracy is improved evidently.

6.2. Comparing to other optimization algorithms

Although the neural network method, simulated annealing algorithm, and simple GA were employed to optimize the Raman-gain spectrum and their optimal results were exciting [10, 11, 14, 15, 19], they only achieved a single optimal solution on the determinate conditions and even their methods may be trapped in local optima of the search space. In our previous reports, the hybrid-GA can offer not only broader signal bandwidth but also several global (or near-global) maximum simultaneously [16]. But, due to the intrinsic shortcoming of GA, its optimization process is time-consuming. In this paper, the novel algorithm, based on the geometry compensation technique and hybrid-GA, increases the optimization efficiency significantly. The former (i.e., geometry compensation technique) enhances the computing speed greatly and the latter (i.e., hybrid-GA) provides multiple global solutions. Compared to the pure hybrid-GA [16], the novel algorithm decreases the CPU time by more than an order of magnitude when the same global or near-global solutions are obtained. For example, it takes about three hours to achieve the results of Table 2 and Fig. 5 if the pure hybrid-GA is implemented, while the run-time is about 15 minutes for the novel algorithm. Therefore, this algorithm has the potential capacity of widely being used in the real-word simulations.

6.3. Comparing to other numerical methods

Recently, to increase the computing speed of designing DFRAs, some novel numerical methods have been proposed by taking into account the fact that the evolutions of pump, signal and noise waves are exponential power approximation. Such novel techniques include the average power analysis [21], predictor-corrector method [27], backward differentiation formulae [2], and so on. Although they are relatively easy to code, unfortunately, all of them adopt a fixed constant step-size. However, a varying step-size algorithm is usually much faster than its constant step-size version, because it contains both rapid variational segments, where smaller steps are required, and slow variational segments, where larger steps are acceptable. Here, a novel automatic step-size method (Eqs. (3) and (4)) is proposed and it concentrates its computational effort only on those variable intervals that need it most (an example is demonstrated in Fig. 6(a)). In comparison with the numerical methods of Refs.[2, 21, 27], the novel method increases the computing speed by more than one or even two orders of magnitude under the same accuracy (the accelerated factor is related to the accuracy). Therefore, the novel fast numerical method is suggested to significantly improve the simulation efficiency of designing DFRAs.

6.4. Comparing to Dormand-Prince method

To prove the merits of the proposed numerical method (i.e., Eqs. (3) and (4)), we give a example for simulating Type-1. In comparison with the classical Dormand-Prince formula (MATLAB has an implementation of Dormand-Prince formula and provides a function called ode45.m), the results are shown in Fig. 6, where the circular symbols represent the steps that are calculated in the automatic adjustment procedure. The local truncation error of each step and the relative error of signals are assumed as 10^-11 and 10^-7, respectively. The detail of Dormand-Prince formula is shown in Appendix-1. Fig. 6(a) is calculated from our method and (b) is obtained from MATLAB. From Fig. 6, one can see that there are only 51 steps for the proposed method instead of 211 steps for the Dormand-Prince method, and the step length of the proposed method is great longer than that of the Dormand-Prince method. Numerical results show that, under the same accuracy, the CPU time of our method is decreased more than 4 times in comparison with the Dormand-Prince method. The key reason why our methods (i.e., Eq. (3)) can offer fast computational efficiency is that Eq. (3) takes into account the fact that the power evolutions of pump, signal and noise waves are the exponential approximation. But, the Dormand-Prince methods are lack.

 

Fig. 6. Pump and signal power evolutions along the fiber: (a) with the proposed method, and (b) with the Dormand-Prince method. The circular symbols in figure represent the points that are calculated in the automatic adjustment procedure.

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6.5. Errors between our model and the rigorous model

The rigorous model equations of DFRAs include such physical terms as spontaneous Raman emission and its temperature dependence, Rayleigh scattering including multiple reflections, ASE, SRS, high-order Stokes generation, and arbitrary interaction between an unlimited number of pumps and signals [1, 22, 29, 30]. Such equations can be obtained from Ref. [22], i.e.,

Although Eq. (5) can be solved by means of multiple shooting algorithms, Eqs. (1) and (2) are employed to optimize the design of DFRAs. Then the gain spectra and noise distribution are optimized independently of each other. The reasons are that, under the same accuracy and parameter set, the CPU time by solving Eqs. (1) and (2) is shortened by ~10 times in comparison with Eq. (5). Of course, the optimal process on the basis of Eqs. (1) and (2) can cause the error. But the error is great little. For example, based on the same parameters in Type-2 of Figs. 2 and 3, the relative errors for the gain and noise distribution are less than 8×10^-3 by comparing the rigorous model Eq. (5) to the simplified model Eqs. (1) and (2). Therefore, under the experimental errors, our optimal results are true. Our results also strongly suggest that the optimal design of DFRAs can be implemented by solving Eqs. (1) and (2) instead of Eq. (5) if the fiber span L is no more than 80 km, and then the gain profile and noise performance of DFRAs can be optimized independently.

However, when the fiber span L and the on-off gain G _on-off are more than 125 km and 25 dB, respectively, the relative error for the noise performance will exceed 10% if using Eq. (2). In this case, the backscattering terms of pumps and signals should be included and the rigorous model Eq. (5) has to be solved to evaluate the gain and noise distribution of DFRAs. Our results are consistent with the reports of Refs. [9, 20, 31].

Since the proposed numerical methods (i.e., Eqs. (3) and (4)) can succeed to fast solve both Eqs. (2) and (5), they can be employed to optimize DFRAs in the case of L>125 km and G _on-off >25 dB. Similarly, the proposed optimal algorithms in Section-3 are also suitable to such case. Of course, it will take more CPU time to design DFRAs because we have to solve Eq. (5) instead of Eqs. (1) and (2).

6.6. Stability

Although the “pure” shooting algorithm has faster computing efficiency [17], it is divergent when the coupled equations of bi-directionally pumped DFRAs with high pump powers are solved. Due to the intrinsic property of GA, the choices of pump power and wavelength are stochastic during the optimizing process, and the candidate solutions may include high pump powers. For example, when the pump wavelengths are the same as the Type-3 of Fig. 2 and each pump power is 350 mW, the “pure” shooting algorithm fails to solve the coupled equation of DFRAs. However, the multiple (e.g., four-subdomain) shooting algorithms can succeed to overcome such difficulty. The detailed iteration process is demonstrated in Fig. 7, where (a)–(f) are the first-, second-, third-, forth-, fifth-and sixth-iteration in the simulations. From Fig. 7, we can see that it is stable for multiple shooting algorithms.

Since the multiple shooting algorithms are more stable and the “pure” shooting algorithm is faster in solving the coupled equations of DFRAs [17], both of them are used in our simulations. At first, the model equations of DFRAs are solved by means of the “pure” shooting algorithm. When the pump power is greater, using such algorithm may give divergent results. In this case, the equations are evaluated by the multiple shooting algorithms instead of the “pure” shooting algorithm. Because shooting algorithms (including the “pure” and multiple shooting algorithms) are beyond the research of this paper, we will not describe them in detail here.

 

Fig. 7. Pump and signal power evolutions along the fiber, where (a)–(f) are the first-, second-, third-, forth-, fifth- and sixth-iteration in the simulations, respectively. All parameters are the same as the Type-3 of Fig. 2 except that each pump power is up to 350 mW.

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7. Conclusions

On the basis of the hybrid-GA, geometry compensation technique and multiple shooting algorithms, a novel algorithm for optimizing the design of DFRAs is proposed. In comparison with pure hybrid-GA [16], the novel algorithm not only can offer multiple global (or near-global) maximums simultaneously but also can increase the computing efficiency by a factor of more than 10. A fast numerical method for solving the model equations of DFRAs is also proposed, which can shorten the amount of run-time more than 4 times by comparison to the classical fast method (i.e., Dormand-Prince method). Compared to the numerical methods with the fixed constant step-size in Refs.[2, 21, 27], our adaptive numerical method can increase the computing speed by more than an order of magnitude. Based on the proposed algorithms, the gain profile, OSNR spectrum and corresponding K_NL for co-, counter- and bi-directional multi-pump Raman amplifiers are optimized and compared in detail. The optimized results show that ① pure forward pumping scheme can provide more than 3-dB OSNR improvement by comparison to pure backward pumping scheme, but the latter has the least K_NL that is less by over a order of magnitude than the former; ② bi-directionally pumping configurations have less ripples of gain and OSNR than pure backward pumping case, whereas the latter has the least total power of input pumps; ③ after optimizing DFRAs, bi-directionally pumping case with appropriate forward power can obviously increase the performances of gain and OSNR and equalize the OSNR tilt at a little cost of nonlinear impact. For example, Type-2 can offer the gain ripple of 0.76 dB, the OSNR ripple of 0.81 dB and the slope of 0.0014 dB/nm for the fitted line of OSNR, instead of 1.02 dB, 2.37 dB and 0.0324 dB/nm for Type-1, respectively, whereas both Type-1 and Type-2 have the approximate same K_NL (i.e., ~0.004 rad). There exist multiple global or near-global solutions for signal gain spectrum (e.g., there are 6 global solutions for Type-2 in our simulations), but different solution has different noise performance. These results reported in this paper can have important applications on the design of DFRAs in practice.

Appendix-1

The Dormand-Prince formula for Eq. (1) is shown as follow.

(6.a)$$Q_1=h·f(z_j,P\left(z_j\right)),$$

(6.b) $$Q_2=h·f\left(z_j+\frac{h}{5},P\left(z_j\right)+\frac{Q_1}{5}\right),$$

(6.c) $$Q_3=h·f\left(z_j+\frac{3h}{10},P\left(z_j\right)+\frac{{3Q}_1}{40}+\frac{{9Q}_2}{40}\right),$$

(6.d) $$Q_4=h·f\left(z_j+\frac{4h}{5},P\left(z_j\right)+\frac{{44Q}_1}{45}-\frac{{56Q}_2}{15}+\frac{{32Q}_3}{9}\right),$$

(6.e) $$Q_5=h·f\left(z_j+\frac{8h}{9},P\left(z_j\right)+\frac{{19372Q}_1}{6561}-\frac{{25360Q}_2}{2187}+\frac{{64448Q}_3}{6561}-\frac{{212Q}_4}{729}\right),$$

(6.f)$$Q_6=h·f\left(z_j+h,P\left(z_j\right)+\frac{{9017Q}_1}{3168}-\frac{{355Q}_2}{33}+\frac{{46732Q}_3}{5247}+\frac{{49Q}_4}{176}-\frac{{5103Q}_5}{18565}\right),$$

(6.g) $$Q_7=h·f\left(z_j+h,P\left(z_j\right)+\frac{{35Q}_1}{384}+\frac{{500Q}_3}{1113}+\frac{{125Q}_4}{192}-\frac{{2187Q}_5}{6784}+\frac{{11Q}_6}{84}\right),$$

(6.h) $$\stackrel{︵}{P\left(z_{j+1}\right)}=P\left(z_j\right)+\frac{{5179Q}_1}{57600}+\frac{{7571Q}_3}{16695}+\frac{{393Q}_4}{640}-\frac{{92097Q}_5}{339200}+\frac{{187Q}_6}{2100}+\frac{Q_7}{40},$$

(6.i) $$P\left(z_{j+1}\right)=P\left(z_j\right)+\frac{{35Q}_1}{384}+\frac{{500Q}_3}{1113}+\frac{{125Q}_4}{192}-\frac{{2187Q}_5}{6784}+\frac{{11Q}_6}{84}.$$

The function f in Eqs. (6) is from Eq. (1.a). The detail implement of Dormand-Prince method can be found from the MATLAB software (i.e., ode45.m). The differences between our method and Dormand-Prince method can be clear seen from the following example. Similar to Eq. (1), an ordinary differential equation dy/dx=y-x/2=y(1-x/(2y)) with the initial condition of y(0)=1.5 is testified, and then the function f(x,y)=y-x/2 and F(x,y)=1-x/(2y). The simulations are shown in Fig. 8, where (a) denotes the automatic step length along the variable x, (b) represents the relative error at each step. The results for Dormand-Prince method in Fig. 8 can be obtained from MATLAB immediately (the MATLAB program can be as: opt=odeset(‘RelTol’,1e-5); f=inline(‘y-x/2’); [x,y]=ode45(f,[0,20],1.5,opt); plot(x(1:end-1),diff(x),‘.-’); figure; plot(x,abs((y-(1/2*x+1/2+exp(x)))./(1/2*x+1/2+exp(x))),‘.-’);). Based on the method of Eq. (3) and the function of ode45.m in MATLAB, it is easy to obtain the curves attached with hollow-circular symbols in Fig. 8. One can see, from Fig. 8, that it is only 30 steps for our proposed method instead of 220 steps for Dormand-Prince method. Therefore, the computing speed of our method increases more than 7 times.

 

Fig. 8. Relationships of automatic step length and relative error with variable x, where (a) for automatic step length and (b) for relative error.

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Appendix-2

The following parts are the source codes of MATLAB for the proposed fast numerical method in Section-4.

function [Pj1,E]=OurMethod(Ffun,zj,Pj,h,flag);

%The meaning for h, E, K1-K7 (i.e., K ₁-K ₇) and zj(i.e., z_j ) is consistent with the text of %paper. Pj, Pj1 and Pjh are P ^±1(z_j ,v_k ), P ^±(z_{j +1} v_k ) and $\stackrel{︵}{P^{}(z_{j+1},v_k)}$, respectively. Ffun %denotes the function F of Eq. (1b).

A=[1/5 3/10 4/5 8/9 1 1];

B=[

1/5 3/40 44/45 19372/6561 9017/3168 35/384

0 9/40 -56/15 -25360/2187 -355/33 0

0 0 32/9 64448/6561 46732/5247 500/1113

0 0 0 -212/729 49/176 125/192

0 0 0 0 -5103/18656 -2187/6784

0 0 0 0 0 11/84

];

C=[35/384 0 500/1113 125/192 -2187/6784 11/84]′;

D=[5179/57600 0 7571/16695 393/640 -92097/339200 187/2100 1/40]′;

K1=h*feval(Ffun,zj,Pj);

K2=h*feval(Ffun,zj+A(1)*h,Pj.*exp(K1*B(1,1)));

K3=h*feval(Ffun,zj+A(2)*h,Pj.*exp([K1 K2]*B(1:2,2)));

K4=h*feval(Ffun,zj+A(3)*h,Pj.*exp([K1 K2 K3]*B(1:3,3)));

K5=h*feval(Ffun,zj+A(4)*h,Pj.*exp([K1 K2 K3 K4]*B(1:4,4)));

K6=h*feval(Ffun,zj+A(5)*h,Pj.*exp([K1 K2 K3 K4 K5]*B(1:5,5)));

K7=h*feval(Ffun,zj+A(6)*h,Pj.*exp([K1 K2 K3 K4 K5 K6]*B(1:6,6)));

Pj1=Pj.*exp([K1 K2 K3 K4 K5 K6]*C);

Pjh=Pj.*exp([K1 K2 K3 K4 K5 K6 K7]*D);

E=norm(Pjh-Pj1);

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