Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA)

• Published in: PloS one Vol. 14; no. 10; p. e0223476
• Main Authors: Shaikh, Amber, Jamal, M Asif, Hanif, Fozia, Khan, M Sadiq Ali, Inayatullah, Syed
• Format: Journal Article
• Language: English
• Published: United States Public Library of Science 10-10-2019; Public Library of Science (PLoS)
• Subjects: Algorithms; Applied mathematics; Biology and Life Sciences; Chebyshev approximation; Computer and Information Sciences; Computer applications; Computer Simulation; Control theory; Delay; Derivatives; Differential equations; Engineering and Technology; Error analysis; Fluid mechanics; Genetic algorithms; Kalman filters; Literature reviews; Mathematical models; Mathematicians; Methods; Models, Theoretical; Neural networks; Neural Networks, Computer; Numerical experiments; Optimization; Partial differential equations; Physical Sciences; Physics; Polynomials; Research and Analysis Methods; Researchers; Simulated annealing; Simulation; Karachi Pakistan; Pakistan
• ISSN: 1932-6203; 1932-6203
• 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.