Investigation of singular ordinary differential equations by a neuroevolutionary approach

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.

Analysis of Multi-Phase Flow Through Porous Media for Imbibition Phenomena by Using the LeNN-WOA-NM Algorithm

“…Since Eq (29) and Eq (36) are twice differentiable. Therefore, first derivative y (η) and second derivative y (η) of Eq (44) are represented by the following equations,…”

“…plugging Eq (44)(45)(46) in governing ordinary differential equations. Eq (29) and Eq (36) will be converted into an equivalent algebraic system of equations that can be solved for unknown parameters ζ n , ψ n and θ n using LeNN-WOA-NM algorithm.…”

“…A neuro-evolutionary algorithm is applied to investigate oscillatory behavior of heart beat [43]. Singular ordinary differential equations are handled by a hybrid of DPSO and artificial neural networks [44]. Fractional differential equations representing the damping materials are analysed by an efficient soft computing algorithm [44].…”

Intelligent Computing Technique Based Supervised Learning for Squeezing Flow Model

“…In the previous research, squeezing flow has been studied using different numerical methods, but stochastic numerical computing that is dealing with artificial intelligence is modified to solve this problem recently. The accurate results provided by stochastic numerical computing have been employed to provide new research in various fields such as fluid mechanics [15][16][17], biological research [18,19], business and finance systems [20,21], models of Panto-graph delay differential systems [22][23][24], plasma science [25], thermodynamics [26], magneto-hydrodynamics [27], solid conductive materials [28], atomic physics [29] and other researches of interest.…”

A Levenberg–Marquardt Backpropagation Neural Network for the Numerical Treatment of Squeezing Flow With Heat Transfer Model

“…Stochastic numerical computing solvers are generated basically by taking advantage of computing-based on artificial neural networks (ANN) modeling and its optimization of the process to solve different problems system of ordinary or partial differential equations. There are many modern applications of stochastic numerical computing solvers in various fields such as nonlinear systems emerging in fluid dynamics [15]- [17], biological mathematics [18]- [20], financial system model [21], neuro-fuzzy model [22], pantograph system [23]- [25] plasma physics [26], fuel catching fire model [27], magneto-hydrodynamics [28] electrical conduction solids [29], and atomic physics [30] are little under significant examples of these solutions. Such facts inspire the authors to explore and incorporate the soft computing architectures as an alternative, precise, and feasible computational approaches for solving the fluid mechanics' systems associated with the squeezing flow system.…”