# Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}and i

_{2}are the constants representing the initial conditions.

- The design of a novel fractional order pantograph singular system (FOPSS) is presented using the suitable derivation process.
- The computing process based on machine learning or soft computing knacks is implemented to solve the novel FOPSS using the applications of the Meyer wavelets based fractional neural network.

- A novel FOPSS is presented using the pantograph differential system (PDS) and fundamental form of the second-order singular model.
- The numerical performance of the novel FOPSS is obtained by using the designed approach FMWs-NN-PSOIPA, which is used to compare the obtained results and to perform the values of the absolute error (AE).
- The Meyer computing solvers via FMWs-NN-PSOIPA is applied to solve three examples based on the novel FOPSS to authenticate the convergence, precision and stability.
- The reliability of the proposed FMWS-NN-PSOIPA is accessible using the statistical procedures in terms of semi-interquartile range (S.I.R), Theil’s inequality coefficient (T.I.C) and variance account for (VAF).

## 2. Construction of the Novel FOPSS

## 3. Methodology: FMWs-NN-PSOIPA

#### 3.1. Objective Function: FMWs-NN

**r**,

**c**and

**b**represent the components of weight vector (

**W**), shown as:

**W**. To assess the weights of FMWs-NN, one may calculate the theory of approximation with the mean squared error terminology to find an error function ${\epsilon}_{Fit}$, given as:

#### 3.2. Optimization of the Network

**X**

_{i}and the velocity is

**V**, while, ${h}_{1}$ and ${h}_{2}$ are acceleration constant factors. A few prominent applications of the PSO are optimal reactive power dispatch [50], fusion of features for detection of brain tumor [51], energy-efficient routing mechanism for mobile sink in wireless sensor networks [52], optimal power flow problems [53], dynamic service composition focusing on quality-of-service evaluations under hybrid networks [54] and enhancing the production of biodiesel from Microalga [55].

_{i}#### 3.3. Performance Indices

## 4. Simulations and Results

^{−1}to 10

^{−3}, 10

^{−2}to 10

^{−3}and 10

^{−1}to 10

^{−3}for Examples 1, 2 and 3, respectively. The convergence is assessed using the FIT, ENSE, TIC and EVAF measures, drawn in Figure 3h for each class of novel FOPSS. It is indicated that the best performance instances of the FIT measure are found around 10

^{−5}–10

^{−6}, 10

^{−4}–10

^{−5}and 10

^{−3}–10

^{−5}for each example of the novel FOPSS. The EVAF performance instances are calculated around 10

^{−4}to 10

^{−5}for each example of the novel FOPSS. The TIC measures’ performance instances are calculated around 10

^{−3}to 10

^{−5}for each example of the novel FOPSS. The ENSE is calculated around 10

^{−4}to 10

^{−5}for each variant of the novel FOPSS.

^{−3}to 10

^{−6}, 10

^{−4}to 10

^{−5}and 10

^{−3}to 10

^{−5}for Examples 1, 2 and 3. TIC lies around 10

^{−3}to 10

^{−5}for Example 1, whereas the other two examples of TIC values are found around 10

^{−3}to 10

^{−5}. The EVAF and ENSE values for each case of the novel FOPSS lie in the ranges of 10

^{−1}to 10

^{−2}and 10

^{−2}to 10

^{−3}, respectively. These best measures, calculated through statistical gages, authenticate the correctness of FMWs-NN-PSOIPA.

^{−3}–10

^{−5}, 10

^{−2}–10

^{−3}, 10

^{−2}–10

^{−3}, 10

^{−2}–10

^{−4}, 10

^{−2}–10

^{−3}and 10

^{−3}–10

^{−5}. In Example 2, the performance lies around 10

^{−2}–10

^{−6}, 10

^{−1}–10

^{−2}, 10

^{−2}–10

^{−4}, 10

^{−2}–10

^{−3}, 10

^{−2}–10

^{−3}and 10

^{−2}–10

^{−4}. Likewise, in Example 3, the performance lies around 10

^{−3}to 10

^{−5}, 10

^{−1}to 10

^{−3}, 10

^{−2}to 10

^{−3}, 10

^{−2}to 10

^{−3}, 10

^{−2}to 10

^{−3}and 10

^{−2}to 10

^{−5}. These calculated consistent and small performance instances of each operative authenticate the accuracy and constancy of FMWs-NN-PSOIPA for solving the novel FOPSS.

^{−5}–10

^{−7}, 10

^{−5}–10

^{−6}, 10

^{−3}–10

^{−4}and 10

^{−2}–10

^{−3}, whereas the S.I.R gages for these measures are found around 10

^{−5}–10

^{−7}, 10

^{−7}–10

^{−8}, 10

^{−4}–10

^{−5}and 10

^{−2}–10

^{−4}to solve the novel FOPSS. The classic global performance validates the clarity of FMWs-NN-PSOIPA.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Sabir, Z.; Raja, M.A.Z.; Guirao, J.L.G.; Shoaib, M. A novel design of fractional Meyer wavelet neural networks with application to the nonlinear singular fractional Lane-Emden systems. Alex. Eng. J.
**2021**, 60, 2641–2659. [Google Scholar] [CrossRef] - Momani, S.; Ibrahim, R.W. On a fractional integral equation of periodic functions involving Weyl–Riesz operator in Banach algebras. J. Math. Anal. Appl.
**2008**, 339, 1210–1219. [Google Scholar] [CrossRef] [Green Version] - Bonilla, B.; Rivero, M.; Trujillo, J.J. On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput.
**2007**, 187, 68–78. [Google Scholar] [CrossRef] - Yu, F. Integrable coupling system of fractional soliton equation hierarchy. Phys. Lett. A
**2009**, 373, 3730–3733. [Google Scholar] [CrossRef] - Diethelm, K.; Ford, N.J. Analysis of fractional differential equations. J. Math. Anal. Appl.
**2002**, 265, 229–248. [Google Scholar] [CrossRef] [Green Version] - Diethelm, K.; Freed, A.D. On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In Scientific Computing in Chemical Engineering II; Springer: Berlin/Heidelberg, Germany, 1999; pp. 217–224. [Google Scholar]
- Zhang, Y.; Sun, H.G.; Stowell, H.H.; Zayernouri, M.; Hansen, S.E. A review of applications of fractional calculus in Earth system dynamics. Chaos Solitons Fractals
**2017**, 102, 29–46. [Google Scholar] [CrossRef] - Evans, R.M.; Katugampola, U.N.; Edwards, D.A. Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem. Comput. Math. Appl.
**2017**, 73, 1346–1362. [Google Scholar] [CrossRef] [Green Version] - Engheia, N. On the role of fractional calculus in electromagnetic theory. IEEE Antennas Propag. Mag.
**1997**, 39, 35–46. [Google Scholar] [CrossRef] - Daou, R.A.Z.; Samarani, F.E.; Yaacoub, C.; Moreau, X. Fractional Derivatives for Edge Detection: Application to Road Obstacles. In Smart Cities Performability, Cognition, & Security; Springer: Cham, Switzerland, 2020; pp. 115–137. [Google Scholar]
- Torvik, P.J.; Bagley, R.L. On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech.
**1984**, 51, 294–298. [Google Scholar] [CrossRef] - Aman, S.; Khan, I.; Ismail, Z.; Salleh, M.Z. Applications of fractional derivatives to nanofluids: Exact and numerical solutions. Math. Model. Nat. Phenom.
**2018**, 13, 2. [Google Scholar] [CrossRef] - Matlob, M.A.; Jamali, Y. The concepts and applications of fractional order differential calculus in modeling of viscoelastic systems: A primer. Crit. Rev. Biomed. Eng.
**2019**, 47, 249–276. [Google Scholar] [CrossRef] - Yang, X.J.; Machado, J.A.T.; Cattani, C.; Gao, F. On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 47, 200–206. [Google Scholar] [CrossRef] - Sabir, Z.; Günerhan, H.; Guirao, J.L.G. On a new model based on third-order nonlinear multisingular functional differential equations. Math. Probl. Eng.
**2020**, 2020, 1683961. [Google Scholar] [CrossRef] [Green Version] - Abdelkawy, M.A.; Sabir, Z.; Guirao, J.L.G.; Saeed, T. Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model. Open Phys.
**2020**, 18, 770–778. [Google Scholar] [CrossRef] - Sabir, Z.; Sakar, M.G.; Yeskindirova, M.; Saldir, O. Numerical investigations to design a novel model based on the fifth order system of Emden–Fowler equations. Theor. Appl. Mech. Lett.
**2020**, 10, 333–342. [Google Scholar] [CrossRef] - Adel, W.; Sabir, Z. Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method. Eur. Phys. J. Plus
**2020**, 135, 427. [Google Scholar] [CrossRef] - Sabir, Z.; Amin, F.; Pohl, D.; Guirao, J.L.G. Intelligence computing approach for solving second order system of Emden–Fowler model. J. Intell. Fuzzy Syst.
**2020**, 38, 7391–7406. [Google Scholar] [CrossRef] - Guirao, J.L.G.; Sabir, Z.; Saeed, T. Design and numerical solutions of a novel third-order nonlinear Emden–Fowler delay differential model. Math. Probl. Eng.
**2020**, 2020, 7359242. [Google Scholar] [CrossRef] - Sabir, Z.; Wahab, H.A.; Umar, M.; Sakar, M.G.; Raja, M.A.Z. Novel design of Morlet wavelet neural network for solving second order Lane-Emden equation. Math. Comput. Simul.
**2020**, 172, 1–14. [Google Scholar] [CrossRef] - Ockendon, J.R.; Tayler, A.B. The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. London A Math. Phys. Sci.
**1971**, 322, 447–468. [Google Scholar] - Wake, G.C.; Cooper, S.; Kim, H.K.; Van-Brunt, B. Functional differential equations for cell-growth models with dispersion. Commun. Appl. Anal.
**2000**, 4, 561–574. [Google Scholar] - Bellen, A.; Guglielmi, N.; Torelli, L. Asymptotic stability properties of θ-methods for the pantograph equation. Appl. Numer. Math.
**1997**, 24, 279–293. [Google Scholar] [CrossRef] - Sinha, A.S.C. Stabilisation of time-varying infinite delay control systems. IEE Proc. D-Control Theory Appl.
**1993**, 140, 60–63. [Google Scholar] [CrossRef] - Ezz-Eldien, S.S.; Wang, Y.; Abdelkawy, M.A.; Zaky, M.A.; Aldraiweesh, A.A.; Machado, J.T. Chebyshev spectral methods for multi-order fractional neutral pantograph equations. Nonlinear Dyn.
**2020**, 100, 3785–3797. [Google Scholar] [CrossRef] - Ezz-Eldien, S.S. On solving systems of multi-pantograph equations via spectral tau method. Appl. Math. Comput.
**2018**, 321, 63–73. [Google Scholar] [CrossRef] - Anakira, N.R.; Jameel, A.; Alomari, A.K.; Saaban, A.; Almahameed, M.; Hashim, I. Approximate solutions of multi-pantograph type delay differential equations using multistage optimal homotopy asymptotic method. J. Math. Fundam. Sci.
**2018**, 50, 221–232. [Google Scholar] [CrossRef] - Isah, A.; Phang, C. A collocation method based on Genocchi operational matrix for solving Emden-Fowler equations. J. Phys. Conf. Ser.
**2020**, 1489, 012022. [Google Scholar] [CrossRef] - Yousefi, S.A.; Noei-Khorshidi, M.; Lotfi, A. Convergence analysis of least squares-Epsilon-Ritz algorithm for solving a general class of pantograph equations. Kragujev. J. Math.
**2018**, 42, 431–439. [Google Scholar] [CrossRef] - AŞI, Ş.Y.B.; Ismailov, N.A. Taylor operation method for solutions of generalized pantograph type delay differential equations. Turk. J. Math.
**2018**, 42, 395–406. [Google Scholar] - Sabir, Z.; Guirao, J.L.G.; Saeed, T. Solving a novel designed second order nonlinear Lane–Emden delay differential model using the heuristic techniques. Appl. Soft Comput.
**2021**, 102, 107105. [Google Scholar] [CrossRef] - Umar, M.; Sabir, Z.; Raja, M.A.Z.; Sánchez, Y.G. A stochastic numerical computing heuristic of SIR nonlinear model based on dengue fever. Results Phys.
**2020**, 19, 103585. [Google Scholar] [CrossRef] - Umar, M.; Sabir, Z.; Raja, M.A.Z. Intelligent computing for numerical treatment of nonlinear prey–predator models. Appl. Soft Comput.
**2019**, 80, 506–524. [Google Scholar] [CrossRef] - Sabir, Z.; Raja, M.A.Z.; Umar, M.; Shoaib, M. Neuro-swarm intelligent computing to solve the second-order singular functional differential model. Eur. Phys. J. Plus
**2020**, 135, 474. [Google Scholar] [CrossRef] - Sabir, Z.; Raja, M.A.Z.; Wahab, H.A.; Shoaib, M.; Gómez-Aguilar, J.F. Integrated neuro-evolution heuristic with sequential quadratic programming for second-order prediction differential models. Numer. Methods Partial. Differ. Equ.
**2020**, 1–17. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Mehmood, J.; Sabir, Z.; Nasab, A.K.; Manzar, M.A. Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput. Appl.
**2019**, 31, 793–812. [Google Scholar] [CrossRef] - Sabir, Z.; Umar, M.; Guirao, J.L.G.; Shoaib, M.; Raja, M.A.Z. Integrated intelligent computing paradigm for nonlinear multi-singular third-order Emden–Fowler equation. Neural Comput. Appl.
**2020**, 33, 3417–3436. [Google Scholar] [CrossRef] - Umar, M.; Sabir, Z.; Raja, M.A.Z.; Shoaib, M.; Gupta, M.; Sánchez, Y.G. A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. Symmetry
**2020**, 12, 1628. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Umar, M.; Sabir, Z.; Khan, J.A.; Baleanu, D. A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Eur. Phys. J. Plus
**2018**, 133, 364. [Google Scholar] [CrossRef] - Sabir, Z.; Raja, M.A.Z.; Le, D.N.; Aly, A.A. A neuro-swarming intelligent heuristic for second-order nonlinear Lane–Emden multi-pantograph delay differential system. Complex Intell. Syst.
**2021**, 1–14. [Google Scholar] [CrossRef] - Sabir, Z.; Baleanu, D.; Raja, M.A.Z.; Guirao, J.L.G. Design of neuro-swarming heuristic solver for multi-pantograph singular delay differential equation. Fractals
**2021**, 29, 2140022. [Google Scholar] [CrossRef] - Sabir, Z.; Raja, M.A.Z.; Arbi, A.; Altamirano, G.C.; Cao, J. Neuro-swarms intelligent computing using Gudermannian kernel for solving a class of second order Lane-Emden singular nonlinear model. AIMS Math
**2021**, 6, 2468–2485. [Google Scholar] [CrossRef] - Khan, I.; Raja, M.A.Z.; Khan, M.A.R.; Shoaib, M.; Islam, S.; Shah, Z. Design of backpropagated intelligent networks for nonlinear second-order Lane–Emden pantograph delay differential systems. Arab. J. Sci. Eng.
**2021**, 1–14. [Google Scholar] [CrossRef] - Nisar, K.; Sabir, Z.; Raja, M.A.Z.; Ibrahim, A.A.A.; Erdogan, F.; Haque, M.R.; Rodrigues, J.J.; Rawat, D.B. Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models. IEEE Access
**2021**, 9, 77845–77862. [Google Scholar] [CrossRef] - Sabir, Z.; Raja, M.A.Z.; Kamal, A.; Guirao, J.L.G.; Le, D.-N.; Saeed, T.; Salama, M. Neuro-swarm heuristic unsign interior-point algorithm to solver a third kind of multi-singular nonlinear systems. Math Biosci. Eng.
**2021**, 18, 5285–5308. [Google Scholar] [CrossRef] [PubMed] - Nisar, K.; Sabir, Z.; Raja, M.A.Z.; Ibrahim, A.; Asri, A.; Rodrigues, J.J.P.C.; Khan, A.S.; Gupta, M.; Kamal, A.; Rawat, D.B. Evolutionary Integrated Heuristic with Gudermannian Neural Networks for Second Kind of Lane–Emden Nonlinear Singular Models. Appl. Sci.
**2021**, 11, 4725. [Google Scholar] [CrossRef] - Sabir, Z.; Nisar, K.; Raja, M.A.Z.; Ibrahim, A.A.B.A.; Rodrigues, J.J.; Al-Basyouni, K.S.; Mahmoud, S.R.; Rawat, D.B. Design of Morlet wavelet neural network for solving the higher order singular nonlinear differential equations. Alex. Eng. J.
**2021**, 60, 5935–5947. [Google Scholar] [CrossRef] - Rosca, D. Wavelet analysis on some surfaces of revolution via area preserving projection. Appl. Comput. Harmon. Anal.
**2011**, 30, 262–272. [Google Scholar] [CrossRef] [Green Version] - Muhammad, Y.; Khan, R.; Ullah, F.; Aslam, M.S.; Raja, M.A.Z. Design of fractional swarming strategy for solution of optimal reactive power dispatch. Neural Comput. Appl.
**2020**, 32, 10501–10518. [Google Scholar] [CrossRef] - Sharif, M.; Amin, J.; Raza, M.; Yasmin, M.; Satapathy, S.C. An integrated design of particle swarm optimization (PSO) with fusion of features for detection of brain tumor. Pattern Recognit. Lett.
**2020**, 129, 150–157. [Google Scholar] [CrossRef] - Tabibi, S.; Ghaffari, A. Energy-efficient routing mechanism for mobile sink in wireless sensor networks using particle swarm optimization algorithm. Wirel. Pers. Commun.
**2019**, 104, 199–216. [Google Scholar] [CrossRef] - Muhammad, Y.; Khan, R.; Raja, M.A.Z.; Ullah, F.; Chaudhary, N.I.; He, Y. Design of fractional swarm intelligent computing with entropy evolution for optimal power flow problems. IEEE Access
**2020**, 8, 111401–111419. [Google Scholar] [CrossRef] - Gao, H.; Zhang, K.; Yang, J.; Wu, F.; Liu, H. Applying improved particle swarm optimization for dynamic service composition focusing on quality of service evaluations under hybrid networks. Int. J. Distrib. Sens. Netw.
**2018**, 14, 1550147718761583. [Google Scholar] [CrossRef] [Green Version] - Wambacq, J.; Ulloa, J.; Lombaert, G.; François, S. Interior-point methods for the phase-field approach to brittle and ductile fracture. Comput. Methods Appl. Mech. Eng.
**2021**, 375, 113612. [Google Scholar] [CrossRef] - Huo, D.; le Blond, S.; Gu, C.; Wei, W.; Yu, D. Optimal operation of interconnected energy hubs by using decomposed hybrid particle swarm and interior-point approach. Int. J. Electr. Power Energy Syst.
**2018**, 95, 36–46. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Ahmed, U.; Zameer, A.; Kiani, A.K.; Chaudhary, N.I. Bio-inspired heuristics hybrid with sequential quadratic programming and interior-point methods for reliable treatment of economic load dispatch problem. Neural Comput. Appl.
**2019**, 31, 447–475. [Google Scholar] [CrossRef] - Theodorakatos, N.P. A nonlinear well-determined model for power system observability using Interior-Point methods. Measurement
**2020**, 152, 107305. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Shah, F.H.; Alaidarous, E.S.; Syam, M.I. Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model. Appl. Soft Comput.
**2017**, 52, 605–629. [Google Scholar] [CrossRef] - Dewasurendra, M.; Vajravelu, K. On the method of inverse mapping for solutions of coupled systems of nonlinear differential equations arising in nanofluid flow, heat and mass transfer. Appl. Math. Nonlinear Sci.
**2018**, 3, 1–14. [Google Scholar] [CrossRef] - Baskonus, H.M.; Bulut, H.; Sulaiman, T.A. New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method. Appl. Math. Nonlinear Sci.
**2019**, 4, 129–138. [Google Scholar] [CrossRef] [Green Version] - İlhan, E.; Kıymaz, I.O. A generalization of truncated M-fractional derivative and applications to fractional differential equations. Appl. Math. Nonlinear Sci.
**2020**, 5, 171–188. [Google Scholar] [CrossRef] [Green Version] - Durur, H.; Tasbozan, O.; Kurt, A. New analytical solutions of conformable time fractional bad and good modified Boussinesq equations. Appl. Math. Nonlinear Sci.
**2020**, 5, 447–454. [Google Scholar] [CrossRef] - Baig, A.Q.; Naeem, M.; Gao, W. Revan and hyper-Revan indices of Octahedral and icosahedral networks. Appl. Math. Nonlinear Sci.
**2018**, 3, 33–40. [Google Scholar] [CrossRef] [Green Version] - Pandey, P.K. A new computational algorithm for the solution of second order initial value problems in ordinary differential equations. Appl. Math. Nonlinear Sci.
**2018**, 3, 167–174. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**Graphical illustrations are provided in (

**a**–

**c**), best weights in (

**d**–

**f**), AE in (

**e**) and performance in (

**f**) for solving the novel FOPSS. (

**a**) Best weights, Example 1; (

**b**) best weights, Example 2; (

**c**) best weights, Example 3; (

**d**) FOPSS results for Examples 1, 2 and 3; ((

**e**) AE performance instances for each class of the novel FOPSS; (

**f**) performance instances for each example of the novel FOPSS.

**Figure 4.**Statistics values through FMWs-NN-PSOIPA for FIT performance with histogram/boxplots for the novel FOPSS. (

**a**) FIT investigations for each example; (

**b**) histograms for 1st example; (

**c**) histograms for 2nd example; (

**d**) histograms for 3rd example; (

**e**) boxplots for 1st example; (

**f**) boxplots for 2nd example; and (

**g**) boxplots for 3rd example.

**Figure 5.**Statistics values through FMWs-NN-PSOIPA for TIC performance with histogram/boxplots for the novel FOPSS. (

**a**) TIC investigations for each example; (

**b**) histograms for 1st example; (

**c**) histograms for 2nd example; (

**d**) histograms for 3rd example; (

**e**) boxplots for 1st example; (

**f**) boxplots for 2nd example; and (

**g**) boxplots for 3rd example.

**Figure 6.**Statistics values through FMWs-NN-PSOIPA for EVAF performance with histogram/boxplots for the novel FOPSS. (

**a**) EVAF investigations for each example; (

**b**) histograms for 1st example; (

**c**) histograms for 2nd example; (

**d**) histograms for 3rd example; (

**e**) boxplots for 1st example; (

**f**) boxplots for 2nd example; and (

**g**) boxplots for 3rd example.

**Figure 7.**Statistics values through FMWs-NN-PSOIPA for ENSE performance with histogram/boxplots for the novel FOPSS. (

**a**) ENSE investigations for each example; (

**b**) histograms for 1st example; (

**c**) histograms for 2nd example; (

**d**) histograms for 3rd example; (

**e**) boxplots for 1st example; (

**f**) boxplots for 2nd example; and (

**g**) boxplots for 3rd example.

Index | Mode | Proposed Outcomes $\mathit{k}(\mathit{\eta})$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | ||

1 | Min | 4 × 10^{−4} | 3 × 10^{−4} | 1 × 10^{−3} | 8 × 10^{−4} | 4 × 10^{−3} | 4 × 10^{−4} | 1 × 10^{−3} | 1 × 10^{−2} | 3 × 10^{−3} | 3 × 10^{−5} |

Max | 4 × 10^{−2} | 3 × 10^{−2} | 3 × 10^{−2} | 3 × 10^{−2} | 4 × 10^{−2} | 6 × 10^{−2} | 8 × 10^{−2} | 9 × 10^{−2} | 7 × 10^{−2} | 5 × 10^{−3} | |

MED | 6 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 4 × 10^{−2} | 5 × 10^{−2} | 5 × 10^{−2} | 3 × 10^{−2} | 1 × 10^{−3} | |

Mean | 4 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 5 × 10^{−2} | 6 × 10^{−2} | 6 × 10^{−2} | 3 × 10^{−2} | 3 × 10^{−4} | |

S.I.R | 6 × 10^{−3} | 6 × 10^{−3} | 6 × 10^{−3} | 7 × 10^{−3} | 8 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−3} | |

STD | 1 × 10^{−3} | 2 × 10^{−3} | 3 × 10^{−3} | 4 × 10^{−3} | 4 × 10^{−3} | 5 × 10^{−3} | 6 × 10^{−3} | 7 × 10^{−3} | 5 × 10^{−3} | 8 × 10^{−5} | |

2 | Min | 3 × 10^{−4} | 8.7 × 10^{−5} | 1 × 10^{−3} | 2 × 10^{−3} | 8 × 10^{−3} | 2 × 10^{−2} | 2 × 10^{−2} | 2 × 10^{−2} | 8 × 10^{−3} | 6 × 10^{−6} |

Max | 7 × 10^{−2} | 8 × 10^{−2} | 8 × 10^{−2} | 8 × 10^{−2} | 7 × 10^{−2} | 7 × 10^{−2} | 8 × 10^{−2} | 1 × 10^{−1} | 9 × 10^{−2} | 1 × 10^{−2} | |

MED | 9 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 4 × 10^{−2} | 6 × 10^{−2} | 6 × 10^{−2} | 4 × 10^{−2} | 1 × 10^{−4} | |

Mean | 7 × 10^{−3} | 6 × 10^{−3} | 1 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 4 × 10^{−2} | 6 × 10^{−2} | 6 × 10^{−2} | 3 × 10^{−2} | 1 × 10^{−3} | |

S.I.R | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 2 × 10^{−2} | 2 × 10^{−3} | |

STD | 2 × 10^{−3} | 6 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 6 × 10^{−4} | |

3 | Min | 1 × 10^{−5} | 4 × 10^{−5} | 2 × 10^{−3} | 3 × 10^{−3} | 1 × 10^{−2} | 9 × 10^{−3} | 6 × 10^{−3} | 3 × 10^{−2} | 1 × 10^{−2} | 7 × 10^{−4} |

Max | 7 × 10^{−2} | 1 × 10^{−1} | 1 × 10^{−1} | 1 × 10^{−1} | 1 × 10^{−1} | 1 × 10^{−1} | 1 × 10^{−1} | 1 × 10^{−1} | 1 × 10^{−1} | 4 × 10^{−3} | |

MED | 9 × 10^{−3} | 1 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 5 × 10^{−2} | 6 × 10^{−2} | 7 × 10^{−2} | 7 × 10^{−2} | 5 × 10^{−2} | 1 × 10^{−3} | |

Mean | 6 × 10^{−3} | 5 × 10^{−3} | 1 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 5 × 10^{−2} | 7 × 10^{−2} | 8 × 10^{−2} | 6 × 10^{−2} | 1 × 10^{−3} | |

S.I.R | 1 × 10^{−2} | 2 × 10^{−2} | 2 × 10^{−2} | 2 × 10^{−2} | 2 × 10^{−2} | 3 × 10^{−2} | 2 × 10^{−2} | 1 × 10^{−2} | 1 × 10^{−2} | 6 × 10^{−4} | |

STD | 2 × 10^{−3} | 9 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−2} | 2 × 10^{−2} | 2 × 10^{−2} | 1 × 10^{−2} | 6 × 10^{−3} | 1 × 10^{−2} | 1 × 10^{−5} |

Index | G.FIT | G.TIC | G.ENSE | G.EVAF | ||||
---|---|---|---|---|---|---|---|---|

Min | SI.R | Min | SIR | MIN | SI.R | Min | SI.R | |

1 | 2.556 × 10^{−6} | 2.285 × 10^{−5} | 7.728 × 10^{−6} | 2.673 × 10^{−7} | 2.291 × 10^{−3} | 3.330 × 10^{−5} | 1.819 × 10^{−2} | 2.636 × 10^{−2} |

2 | 5.527 × 10^{−7} | 3.006 × 10^{−5} | 1.007 × 10^{−5} | 3.272 × 10^{−7} | 3.844 × 10^{−3} | 6.650 × 10^{−5} | 2.938 × 10^{−2} | 5.920 × 10^{−4} |

3 | 5.399 × 10^{−5} | 7.898 × 10^{−7} | 1.130 × 10^{−5} | 3.491 × 10^{−8} | 2.291 × 10^{−4} | 3.330 × 10^{−4} | 7.908 × 10^{−3} | 7.100 × 10^{−2} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sabir, Z.; Raja, M.A.Z.; Guirao, J.L.G.; Saeed, T.
Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System. *Fractal Fract.* **2021**, *5*, 277.
https://doi.org/10.3390/fractalfract5040277

**AMA Style**

Sabir Z, Raja MAZ, Guirao JLG, Saeed T.
Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System. *Fractal and Fractional*. 2021; 5(4):277.
https://doi.org/10.3390/fractalfract5040277

**Chicago/Turabian Style**

Sabir, Zulqurnain, Muhammad Asif Zahoor Raja, Juan L. G. Guirao, and Tareq Saeed.
2021. "Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System" *Fractal and Fractional* 5, no. 4: 277.
https://doi.org/10.3390/fractalfract5040277