An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch’s problem

Yadav, Neha; Yadav, Anupam; Kumar, Manoj; Kim, Joong Hoon

doi:10.1007/s00521-015-2046-1

Cited by 41 publications

(12 citation statements)

References 25 publications

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“…Most of the real-world problems are mathematically modeled in the form of differential equations containing the derivative of integer and/ or fractional order. ANNs are frequently used to suggest approximate solutions for such problems [36][37][38][39][40][41].…”

Section: Ann Based Approximationmentioning

confidence: 99%

“…A detailed discussion of neural networks is given in [36,37,41]. An activation function uðxÞ ¼ 1 1þe À x also known as Log-sigmoid mapping function is used to train the unknown weights.…”

Section: Plos Onementioning

confidence: 99%

“…The mathematical form of an approximate solution for BVPs with doubly singularities is suggested by feed-forward ANN as given in Eq (4) , ŷ ( x ) represents the approximate solution, x is the independent variable, γ i , β i and w i are the unknown weights, and Eq (5) expresses the n th derivative of this approximate solution. A detailed discussion of neural networks is given in [ 36 , 37 , 41 ]. An activation function also known as Log-sigmoid mapping function is used to train the unknown weights.…”

Section: The Hybrid Ann and Fo-dpso Approachmentioning

confidence: 99%

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Investigation of singular ordinary differential equations by a neuroevolutionary approach

et al. 2020

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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.

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Section: Ann Based Approximationmentioning

confidence: 99%

Section: Plos Onementioning

confidence: 99%

Section: The Hybrid Ann and Fo-dpso Approachmentioning

confidence: 99%

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Investigation of singular ordinary differential equations by a neuroevolutionary approach

et al. 2020

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“…Numerical optimization techniques like genetic algorithm [38], [39], different variants of particle swarm optimization (PSO) [40]- [42], interior point algorithm [43], evolutionary programming [44] and others [45], [46] are used to solve mathematical models involving ODEs and PDEs. In [41], a variant of PSO is designed and applied to solve nonlinear ODEs. Soft computing techniques are applied to generate visible videos from raw footage, which has many applications, like, video compression, movie production, slow-motion filming, video surveillance, and forensic analysis [47].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Temperature Profiles in Longitudinal Fin Designs by a Novel Neuroevolutionary Approach

et al. 2020

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Real application problems in physics, engineering, economics, and other disciplines are often modeled as differential equations. Classical numerical techniques are computationally expensive when we require solutions to our mathematical problems with no prior information. Hence, researchers are more interested in developing numerical methods that can obtain better solutions with fewer efforts and computational time. Heuristic algorithms are considered suitable candidates for such type of problems. In this research, we have developed a new neuroevolutionary algorithm that combines the power of feed-forward artificial neural networks (ANNs) and a modern metaheuristic, the Symbiotic Organism Search (SOS) algorithm. With our new approach, we have analyzed the simultaneous surface convection and radiation during heat transfer in different models of fins/ heat exchangers. Longitudinal fins are considered with concave parabolic, rectangular and trapezoidal shapes. We have analyzed our problem in two scenarios and six sub-cases. Our solutions are of high quality, with minimum residual errors in all cases. We have established the quality of our results by calculating values of different performance indicators like Root-mean-square error (RMSE), Absolute error (AE), Generational distance (GD), Mean absolute deviation (MAD), Nash-Sutcliffe efficiency (NSE), Error in Nash-Sutcliffe efficiency (ENSE). Statistical and graphical analysis of our results suggests that our approach is suitable for handling real application problems. We have compared our results with state-of-the-art results, and the outcome of our analysis points to the superiority of our approach. INDEX TERMS Approximate solutions, Artificial neural networks, Heat transfer analyses, Longitudinal fins, Metaheuristics, Symbiotic organism search optimizer.

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“…In Ahmad and Bilal (2014), Blasius equation has been solved using neural networks while in Biglari et al (2013), it is solved using neural networks hybridised with gravitational search algorithm. In literature several other attempts has been made to solve differential equations in general using PSO (Babaei, 2013;Yadav et al, 2015). However, there does not exists literature related to solving differential equation using membrane algorithms based on PSO.…”

Section: Introductionmentioning

confidence: 99%

Cell-like P-systems coupled with rules of particle swarm optimisation to solve Blasius differential equation

Singh

Deep

2016

IJSI

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Abstract:In this paper, a non-traditional approach is presented to numerically solve the well-known differential equation, namely Blasius equation, by modelling it as a nonlinear optimisation problem, in which parameters of the nonlinear approximation function satisfying the boundary conditions are identified, by minimising the error. The optimisation methodology involves the use of cell-like P-systems, in which hierarchical membrane structures use the evolution rules of particle swarm optimisation. A comparative implementation of five such P-systems is used to solve the Blasius equation. It is concluded that all the five algorithms provide solutions close to the exact solution and well within the specified lower and upper bounds. It is suggested that the presented approach can be easily extended to solve a wide range of similar problems.

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An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch’s problem

Cited by 41 publications

References 25 publications

Investigation of singular ordinary differential equations by a neuroevolutionary approach

Investigation of singular ordinary differential equations by a neuroevolutionary approach

Analysis of Temperature Profiles in Longitudinal Fin Designs by a Novel Neuroevolutionary Approach

Cell-like P-systems coupled with rules of particle swarm optimisation to solve Blasius differential equation

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