Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA)
Autoři:
Amber Shaikh aff001; M. Asif Jamal aff002; Fozia Hanif aff003; M. Sadiq Ali Khan aff004; Syed Inayatullah aff005
Působiště autorů:
Department of Humanities and Sciences, National University of Computer and Emerging Sciences, Karachi, Pakistan
aff001; Department of Basic Sciences Federal Urdu University of Art, Science and technology Karachi & Cadet College, Karachi, Pakistan
aff002; Department of Mathematics, University of Karachi, Karachi, Pakistan
aff003; Department of Computer Sciences, University of Karachi, Karachi, Pakistan
aff004; Department of Mathematics, University of Karachi, Karachi, Pakistan
aff005
Vyšlo v časopise:
PLoS ONE 14(10)
Kategorie:
Research Article
doi:
https://doi.org/10.1371/journal.pone.0223476
Souhrn
To enrich any model and its dynamics introduction of delay is useful, that models a precise description of real-life phenomena. Differential equations in which current time derivatives count on the solution and its derivatives at a prior time are known as delay differential equations (DDEs). In this study, we are introducing new techniques for finding the numerical solution of fractional delay differential equations (FDDEs) based on the application of neural minimization (NM) by utilizing Chebyshev simulated annealing neural network (ChSANN) and Legendre simulated annealing neural network (LSANN). The main purpose of using Chebyshev and Legendre polynomials, along with simulated annealing (SA), is to reduce mean square error (MSE) that leads to more accurate numerical approximations. This study provides the application of ChSANN and LSANN for solving DDEs and FDDEs. Proposed schemes can be effortlessly executed by using Mathematica or MATLAB software to get explicit solutions. Computational outcomes are depicted, for various numerical experiments, numerically and graphically with error analysis to demonstrate the accuracy and efficiency of the methods.
Klíčová slova:
Algorithms – Control theory – Differential equations – Mathematical functions – Neural networks – Optimization – Polynomials – Simulated annealing
Zdroje
1. Li Z, Chen D, Zhu J, Liu Y. Nonlinear dynamics of fractional order Duffing system. Chaos, Solitons & Fractals. 2015 Dec 1;81:111–6.
2. Pourdehi S, Azami A, Shabaninia F. Fuzzy Kalman-type filter for interval fractional-order systems with finite-step auto-correlated process noises. Neurocomputing. 2015 Jul 2;159:44–9.
3. Boulkroune A, Bouzeriba A, Bouden T. Fuzzy generalized projective synchronization of incommensurate fractional-order chaotic systems. Neurocomputing. 2016 Jan 15;173:606–14.
4. Coronel-Escamilla A, Torres F, Gomez-Aguilar JF, Escobar-Jimenez RF, Guerrero-Ramírez GV. On the trajectory tracking control for an SCARA robot manipulator in a fractional model driven by induction motors with PSO tuning. Multibody System Dynamics. 2018 Jul 15;43(3):257–77.
5. Zhang Y, Mei J, Zhang X. Symmetry properties and explicit solutions of some nonlinear differential and fractional equations. Applied Mathematics and Computation. 2018 Nov 15;337:408–18.
6. Zhang Y, Zhao Z. Lie symmetry analysis, Lie-Bäcklund symmetries, explicit solutions, and conservation laws of Drinfeld-Sokolov-Wilson system. Boundary Value Problems. 2017 Dec 1;2017(1):154.
7. Yang XJ, Gao F, Srivastava HM. Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Computers & Mathematics with Applications. 2017 Jan 15;73(2):203–10.
8. Yang XJ, Machado JT, Baleanu D. Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals. 2017 Aug;25(04):1740006.
9. Atangana A, Gómez-Aguilar JF. Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena. The European Physical Journal Plus. 2018 Apr;133:1–22.
10. Li Y, Zhao W. Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematics and Computation. 2010 Jun 15;216(8):2276–85.
11. Yuanlu L. Solving a nonlinear fractional differential equation using Chebyshev wavelets. Communications in Nonlinear Science and Numerical Simulation. 2010 Sep 1;15(9):2284–92.
12. Odibat Z, Momani S. Numerical methods for nonlinear partial differential equations of fractional order. Applied Mathematical Modelling. 2008 Jan 1;32(1):28–39.
13. Momani S, Odibat Z. Numerical approach to differential equations of fractional order. Journal of Computational and Applied Mathematics. 2007 Oct 1;207(1):96–110.
14. El-Wakil SA, Elhanbaly A, Abdou MA. Adomian decomposition method for solving fractional nonlinear differential equations. Applied Mathematics and Computation. 2006 Nov 1;182(1):313–24.
15. Hosseinnia SH, Ranjbar A, Momani S. Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers &Mathematics with Applications. 2008 Dec 1;56(12):3138–49.
16. Dhaigude DB, Birajdar GA. Numerical solution of system of fractional partial differential equations by discrete Adomian decomposition method. J. Frac. Cal. Appl. 2012 Jul;3(12):1–1.
17. Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method. Chaos, Solitons & Fractals. 2007 Dec 1;34(5):1473–81.
18. Arikoglu A, Ozkol I. Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons & Fractals. 2009 Apr 30;40(2):521–9.
19. Darania P, Ebadian A. A method for the numerical solution of the integro-differential equations. Applied Mathematics and Computation. 2007 May 1;188(1):657–68.
20. Ertürk VS, Momani S. Solving systems of fractional differential equations using differential transform method. Journal of Computational and Applied Mathematics. 2008 May 15;215(1):142–51.
21. Erturk VS, Momani S, Odibat Z. Application of generalized differential transform method to multi-order fractional differential equations. Communications in Nonlinear Science and Numerical Simulation. 2008 Oct 1;13(8):1642–54.
22. Odibat ZM, Shawagfeh NT. Generalized Taylor’s formula. Applied Mathematics and Computation. 2007 Mar 1;186(1):286–93.
23. Aarts LP, Van Der Veer P. Neural network method for solving partial differential equations. Neural Processing Letters. 2001 Dec 1;14(3):261–71.
24. Meade AJ Jr, Fernandez AA. The numerical solution of linear ordinary differential equations by feedforward neural networks. Mathematical and Computer Modelling. 1994 Jun 1;19(12):1–25.
25. Parisi DR, Mariani MC, Laborde MA. Solving differential equations with unsupervised neural networks. Chemical Engineering and Processing: Process Intensification. 2003 Aug 1;42(8–9):715–21.
26. Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks. 1998 Sep;9(5):987–1000. doi: 10.1109/72.712178 18255782
27. Malek A, Beidokhti RS. Numerical solution for high order differential equations using a hybrid neural network—optimization method. Applied Mathematics and Computation. 2006 Dec 1;183(1):260–71.
28. Zúñiga-Aguilar CJ, Romero-Ugalde HM, Gómez-Aguilar JF, Escobar-Jiménez RF, Valtierra-Rodríguez M. Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks. Chaos, Solitons & Fractals. 2017 Oct 1;103:382–403.
29. Davis LC. Modifications of the optimal velocity traffic model to include delay due to driver reaction time. Physica A: Statistical Mechanics and its Applications. 2003 Mar 1;319:557–67.
30. Epstein IR, Luo Y. Differential delay equations in chemical kinetics. Nonlinear models: The cross‐shaped phase diagram and the Oregonator. The Journal of chemical physics. 1991 Jul 1;95(1):244–54.
31. Kuang Y, editor. Delay differential equations: with applications in population dynamics. Academic Press; 1993 Mar 5.
32. Benchohra M, Henderson J, Ntouyas SK, Ouahab A. Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications. 2008 Feb 15;338(2):1340–50.
33. Zúñiga-Aguilar CJ, Coronel-Escamilla A, Gómez-Aguilar JF, Alvarado-Martínez VM, Romero-Ugalde HM. New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks. The European Physical Journal Plus. 2018 Feb 1;133(2):75.
34. Henderson J, Ouahab A. Fractional functional differential inclusions with finite delay. Nonlinear Analysis: Theory, Methods & Applications. 2009 Mar 1;70(5):2091–105.
35. Maraaba TA, Jarad F, Baleanu D. On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Science in China Series A: Mathematics. 2008 Oct 1;51(10):1775–86.
36. Maraaba T, Baleanu D, Jarad F. Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. Journal of Mathematical Physics. 2008 Aug;49(8):083507.
37. Khan NA, Shaikh A. A smart amalgamation of spectral neural algorithm for nonlinear Lane-Emden equations with simulated annealing. Journal of Artificial Intelligence and Soft Computing Research. 2017 Jul 1;7(3):215–24.
38. Khan NA, Shaikh A, Sultan F, Ara A. Numerical Simulation Using Artificial Neural Network on Fractional Differential Equations. In Numerical Simulation-From Brain Imaging to Turbulent Flows 2016. InTech.
39. Yang XJ, Baleanu D, Srivastava HM. Local fractional integral transforms and their applications. Academic Press; 2015 Oct 22.
40. Pao YH, Takefuji Y. Functional-link net computing: theory, system architecture, and functionalities. Computer. 1992 May;25(5):76–9.
41. Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks. 1998 Sep;9(5):987–1000. doi: 10.1109/72.712178 18255782
42. Ledesma S, Aviña G, Sanchez R. Practical considerations for simulated annealing implementation. InSimulated Annealing 2008. InTech.
43. Saeed U. Radial basis function networks for delay differential equation. Arabian Journal of Mathematics. 2016 Sep 1;5(3):139–44.
44. Saeed U. Hermite wavelet method for fractional delay differential equations. Journal of Difference Equations. 2014;2014.
45. Iqbal MA, Saeed U, Mohyud-Din ST. Modified Laguerre wavelets method for delay differential equations of fractional-order. Egypt. J. Basic Appl. Sci. 2015 Mar 1;2:50.
Článek vyšel v časopise
PLOS One
2019 Číslo 10
- Tisícileté topoly, mokří psi, stárnoucí kočky a ospalé octomilky – „jednohubky“ z výzkumu 2024/41
- Jaké jsou aktuální trendy v léčbě karcinomu slinivky?
- Může hubnutí souviset s vyšším rizikem nádorových onemocnění?
- Menstruační krev má značný diagnostický potenciál, mimo jiné u diabetu
- Metamizol jako analgetikum první volby: kdy, pro koho, jak a proč?
Nejčtenější v tomto čísle
- Correction: Low dose naltrexone: Effects on medication in rheumatoid and seropositive arthritis. A nationwide register-based controlled quasi-experimental before-after study
- Combining CDK4/6 inhibitors ribociclib and palbociclib with cytotoxic agents does not enhance cytotoxicity
- Experimentally validated simulation of coronary stents considering different dogboning ratios and asymmetric stent positioning
- Prevalence of pectus excavatum (PE), pectus carinatum (PC), tracheal hypoplasia, thoracic spine deformities and lateral heart displacement in thoracic radiographs of screw-tailed brachycephalic dogs
Zvyšte si kvalifikaci online z pohodlí domova
Všechny kurzy