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: 10.1371/journal.pone.0223476


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


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