Abstract

In this research, we have investigated doubly singular ordinary differential equations and a real application problem of studying the temperature profile in a porous fin model. We have suggested a novel soft computing strategy for the training of unknown weights involved in the feed-forward artificial neural networks (ANNs). Our neuroevolutionary approach is used to suggest approximate solutions to a highly nonlinear doubly singular type of differential equations. We have considered a real application from thermodynamics, which analyses the temperature profile in porous fins. For this purpose, we have used the optimizer, namely, the fractional-order particle swarm optimization technique (FO-DPSO), to minimize errors in solutions through fitness functions. ANNs are used to design the approximate series of solutions to problems considered in this paper. We find the values of unknown weights such that the approximate solutions to these problems have a minimum residual error. For global search in the domain, we have initialized FO-DPSO with random solutions, and it collects best so far solutions in each generation/ iteration. In the second phase, we have fine-tuned our algorithm by initializing FO-DPSO with the collection of best so far solutions. It is graphically illustrated that this strategy is very efficient in terms of convergence and minimum mean squared error in its best solutions. We can use this strategy for the higher-order system of differential equations modeling different important real applications.

Highlights

  • Real-world problems which are modeled as a singular boundary value problem (BVP) of ordinary differential equations are often hard to solve

  • Results got by fractional-order particle swarm optimization technique (FO-DPSO) are compared with exact solutions, Genetic algorithm, its variant GA-SQP and are presented in Tables 1, 2, 3, 4, 5, 6, and 7 with step sizes h = 0.05 and 0.2, for problems 1, 2, 3 and 4

  • We present a new soft computing approach that combines artificial neural networks with a fractional-order particle swarm optimization (FO-DPSO) algorithm

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Summary

Introduction

Real-world problems which are modeled as a singular boundary value problem (BVP) of ordinary differential equations are often hard to solve. We have understood the singular doubly boundary value problem and based on this understanding we have developed our proposed soft computing approach to get better numerical solutions of these problems. In the recent couple of years, alternate approaches based on artificial neural networks combined with heuristics and meta-heuristic are extensively developed to solve non-linear differential equations. In [33], two techniques namely GA and SQP are combined to tackle the doubly non-linear singular differential equations This combined algorithm takes more time and is computationally expensive. Our novel soft computing approach is used to solve non-linear doubly singular differential equations, see Fig 1.

The hybrid ANN and FO-DPSO approach
ANN based approximation
Search method of FO-DPSO
Problem 1
Problem 2
Problem 3
Statistical analysis
Conclusion

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