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Framework for treating non-Linear multi-term fractional differential equations with reasonable spectrum of two-point boundary conditions

Department of Mathematics, University of Karachi, Karachi 75270, Pakistan

Special Issues: Initial and Boundary Value Problems for Differential Equations

In this work, a novel bio-inspired meta-heuristic framework is presented for the assessment of linear and non-linear multi-term fractional differential equations (MFDEs) based on the idea of residual power series method (RPSM) in amalgamation with the bat algorithm (BATA) which, mimics the echolocation behavior of the foraging bats. The bat-inspired based methodology has been implemented to solve MFDEs with different possible variants of two point boundary conditions. The BATA is utilized in the recommended new residual optimization technique (NROT) for the minimization of the energy function attained by the spirit of RPSM. Moreover, to ratify the correctness and accuracy of the deliberated technique the results are evaluated by using two other meta-heuristic optimization techniques i.e., differential evolution algorithm (DEA) and the accelerated particle swarm optimization algorithm (APSOA) for the learning of unknown weights in the derived fitness function. The accuracy and competency of the NROT is validated by comparing the BATA computed results with the exact solution and the corresponding data acquired by the DEA and APSOA. Furthermore, detailed performance analysis is performed through statistical inference based on the large number of independent runs.
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1. A. B. Salati, M. Shamsi and D. F. M. Torres, Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems, Commun. Nonlinear Sci., 67 (2019), 334-350.

2. V. Gallican and R. Brenner, Homogenization estimates for the effective response of fractional viscoelastic particulate composites, Continuum Mech. Therm., 31 (2018), 823-840.

3. R. Ghaffari and F. Ghoreishi, Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations, Appl. Numer. Math., 137 (2019), 62-79.

4. M. Alquran, K. Al-Khaled, S. Sivasundaram, et al. Mathematical and numerical study of existence of bifurcations of the generalized fractional Burgers-Huxley equation, Nonlinear Stud., 24 (2017), 235-244.

5. Y. Povstenko and T. Kyrylych, Time-fractional heat conduction in a plane with two external half-infinite line slits under heat flux loading, Symmetry, 11 (2019), 689.

6. R. L. Bagley and J. Torvik, Fractional calculus-a different approach to the analysis of viscoelastically damped structures, AIAA J., 21 (1983), 741-748.

7. P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.

8. O. A. Arqub and M. Al-Smadi, Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlevé equations in Hilbert space, Chaos, Solitons Fractals, 117 (2018), 161-167.    

9. M. Esmaeilbeigi, M. Paripour and G. Garmanjani, Approximate solution of the fuzzy fractional Bagley-Torvik equation by the RBF collocation method, Comput. Meth. Differ. Eq., 6 (2018), 186–214.

10. M. Uddin and S. Ahmad, On the numerical solution of Bagley-Torvik equation via the Laplace transform. Tbilisi Math. J., 10 (2017), 279-284.

11. M. G. Sakar, O. Saldır and A. Akgül, A novel technique for fractional Bagley-Torvik equation, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 2018, 1-7.

12. J. I. Fister, X. S.Yang, I. Fister, et al. A brief review of nature-inspired algorithms for optimization, 2013, arXiv preprint arXiv:1307.4186.

13. P. Sindhuja, P. Ramamoorthy and M. S. Kumar, A brief survey on nature Inspired algorithms: Clever algorithms for optimization, Asian J. Comput. Sci. Tech., 7 (2018), 27-32.

14. J. C. Bansal, P. K. Singh and N. R. Pal, Evolutionary and Swarm Intelligence Algorithms, Springer, 2019.

15. N. A. Khan, T. Hameed, O. A. Razzaq, et al. Intelligent computing for duffing-harmonic oscillator equation via the bio-evolutionary optimization algorithm, J. Low Freq. Noise V. A., 2018, 1461348418819408.

16. M. Mareli and B. Twala, An adaptive cuckoo search algorithm for optimisation, Appl. Comput. inf., 14 (2018), 107-115.

17. M. Nouiri, A. Bekrar, A. Jemai, et al. An effective and distributed particle swarm optimization algorithm for flexible job-shop scheduling problem, J. Intell. Manuf., 29 (2018), 603-615.    

18. I. Rahman, P. M. Vasant, B. S. M. Singh, et al. On the performance of accelerated particle swarm optimization for charging plug-in hybrid electric vehicles, Alex. Eng. J., 55 (2016), 419-426.

19. M. Dorigo and T. Stützle, Ant colony optimization: Overview and recent advances, In: Handbook of Metaheuristics, Springer, 2019, 311-351.

20. Z. Cui, F. Li and W. Zhang, Bat algorithm with principal component analysis, Int. J. Mach. Learn. Cyb., 10 (2019), 603-622.

21. T. Jayabarathi, T. Raghunathan and A. Gandomi, The bat algorithm, variants and some practical engineering applications: A review, In: Nature-Inspired Algorithms and Applied Optimization, Springer, 2018, 313-330.

22. M. Rahmani, A. Ghanbari and M. M. Ettefagh, A novel adaptive neural network integral sliding-mode control of a biped robot using bat algorithm, J. Vib. Control, 24 (2018), 2045-2060.    

23. S. Chatterjee, S. Sarkar, N. Dey, et al. Hybrid non-dominated sorting genetic algorithm: II-neural network approach, In: Advancements in Applied Metaheuristic Computing, IGI Global, 2018, 264-286.

24. A. Ara, N. A.Khan, F. Naz, et al. Numerical simulation for Jeffery-Hamel flow and heat transfer of micropolar fluid based on differential evolution algorithm, AIP Adv., 8 (2018), 015201.

25. A. Ara, N. A. Khan, O. A. Razzaq, et al. Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling, Adv. Differ. Eq., 2018 (2018), 8.

26. Z. A. A. Alyasseri, A. T. Khader, M. A. Al-Betar, et al. Variants of the flower pollination algorithm: A review, In: Nature-Inspired Algorithms and Applied Optimization, Springer, 2018, 91-118.

27. I. A. Carvalho, D. G. da Rocha, J. G. R. Silva, et al. Study of parameter sensitivity on bat algorithm, In: International Conference on Computational Science and Its Applications, Springer, 2017, 494-508.

28. O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31-52.

29. A. El-Ajou, O. Arqub, Z. Zhour, et al. New results on fractional power series: Theories and applications, Entropy, 15 (2013), 5305-5323.

30. M. Ali, I. Jaradat and M. Alquran, New computational method for solving fractional riccati equation, J. Math. Comput. Sci., 17 (2017), 106-114.

31. M. Alquran, H. Jaradat and M. I. Syam, Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlinear Dynam., 90 (2017), 2525-2529.

32. J. Zhang, Z. Wei, L Yong, et al. Analytical solution for the time fractional BBM-Burger equation by using modified residual power series method, Complexity, 2018 (2018), 2891373.

33. M. Alquran and I. Jaradat, A novel scheme for solving Caputo time-fractional nonlinear equations: Theory and application, Nonlinear Dynam., 91 (2018), 2389-2395.

34. I. Jaradat, M. Al-Dolat, K. Al-Zoubi, et al. Theory and applications of a more general form for fractional power series expansion, Chaos, Solitons Fractals, 108 (2018), 107-110.

35. N. A. Khan, O. A. Razzaq, T. Hameed, et al., Numerical scheme for global optimization of fractional optimal control problem with boundary conditions, In. J. Innov. Comput. Inform. Control, 13 (2017), 1669-1679.

36. R. Bharathi and M. Tech, Study of comparison between bat algorithm, particle swarm optimization (PSO), grey wolf optimization (GWO) for user's bank loan and their related due history, In. J. Sci. Res. Comput. Sci., Eng. Inform. Tech., 3 (2018), 2456-3307.

37. X. Meng, X. Gao and Y. Liu, A novel hybrid bat algorithm with differential evolution strategy for constrained optimization. In. J. Hybrid Inform. Tech., 8 (2015), 383-396.

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