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

Amber Shaikh,Fozia Hanif,M Sadiq Ali Khan,M Asif Jamal,Syed Inayatullah,Bedreddine Ainseba

doi:10.1371/journal.pone.0223476

Abstract

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.

Highlights

In the old days, fractional calculus was only used by pure mathematicians due to its imperceptible applications at that time
It shows that accuracy of both the methods is inversely proportional to the value of mean square error (MSE) and it can be noticed that change of polynomials in both methods has strongly influenced the learning of network adaptive coefficient (NAC) by simulated annealing (SA) algorithm that can be witnessed from Table 1
In above study we have developed two methods Chebyshev simulated annealing neural network (ChSANN) and Legendre simulated annealing neural network (LSANN) for simulation of fractional delay differential equation

Summary

Introduction

Fractional calculus was only used by pure mathematicians due to its imperceptible applications at that time. When mathematicians were trying to implement fractional calculus in modeling of physical phenomena applicability of this marvelous tool was successfully revealed. In recent years, fractional derivatives have been used in many phenomena in electromagnetic theory, fluid mechanics, viscoelasticity, circuit theory, control theory, biology, atmospheric physics, etc. Many real-world problems can be accurately modeled by fractional differential equations (FDEs) such as damping laws, fluid mechanics, rheology, physics, mathematical biology, diffusion processes, electrochemistry, and so on.

Methods

Discussion

Conclusion