Design of Morlet wavelet neural network for solving the higher order singular nonlinear differential equations
TL;DR: The Morlet wavelet neural networks is applied to discretize the higher order singular nonlinear differential equations to express the activation function using the mean square error to check the significance, efficacy and consistency of the designed MWNNs using the GA-IPM.
Abstract: The aim of this study is to present the numerical solutions of the higher order singular nonlinear differential equations using an advanced intelligent computational approach by manipulating the Morlet wavelet (MW) neural networks (NNs), global approach as genetic algorithm (GA) and quick local search approach as interior-point method (IPM), i.e., GA-IPM. MWNNs is applied to discretize the higher order singular nonlinear differential equations to express the activation function using the mean square error. The performance of the designed MWNNs using the GA-IPM is observed to solve three different variants based on the higher order singular nonlinear differential model to check the significance, efficacy and consistency of the designed MWNNs using the GA-IPM. Furthermore, statistical performances are provided to check the precision, accuracy and convergence of the present approach.
Citations
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TL;DR: The verification, perfection and authentication of the singular fractional order pantograph model using fractional Meyer computing solver is observed for different cases through comparative studies from the available exact solutions which endorsed its robustness, convergence and stability.
Abstract: The aim of this study is to design a singular fractional order pantograph differential model by using the typical form of the Lane-Emden model. The necessary details of the singular-point, fractional order and shape factor of the designed model are also provided. The numerical solutions of the designed model have been presented using the combination of the fractional Meyer wavelet (FMW) neural networks (NNs) modeling and optimization of global search with genetic algorithm (GA) supported with local search of sequential quadratic programming (SQP), i.e., FMWNN-GASQP. The strength of FMWNN is employed to design an objective function using the differential model along with its initial conditions of the singular fractional order pantograph model. The optimization of this objective function is performed using the integrated competence of GA-SQP. The verification, perfection and authentication of the singular fractional order pantograph model using fractional Meyer computing solver is observed for different cases through comparative studies from the available exact solutions which endorsed its robustness, convergence and stability. Moreover, the statistics observation with necessary explanations further authenticate the performance of the FMWNN-GASQP in terms of accuracy and reliability.
36 citations
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TL;DR: In this article , the authors investigated the GNNs using the optimization procedures of genetic algorithm and active set approach (GA-ASA) to solve the three-species food chain nonlinear model.
Abstract: The present study is to investigate the Gudermannian neural networks (GNNs) using the optimization procedures of genetic algorithm and active-set approach (GA-ASA) to solve the three-species food chain nonlinear model. The three-species food chain nonlinear model is dependent upon the prey populations, top-predator, and specialist predator. The design of an error-based fitness function is presented using the sense of the three-species food chain nonlinear model and its initial conditions. The numerical results of the model have been obtained by exploiting the GNN-GA-ASA. The obtained results through the GNN-GA-ASA have been compared with the Runge–Kutta method to substantiate the correctness of the designed approach. The reliability, efficacy and authenticity of the proposed GNN-GA-ASA are examined through different statistical measures based on single and multiple neurons for solving the three-species food chain nonlinear model.
26 citations
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TL;DR: The explanations via the statistical measures validate the value of the designed stochastic solver FMW-NN-GAIPA, which is designed to characterize the novel model in the sagacity of mean squared error of objective function.
Abstract: The present study is related to design a novel multi-fractional multi-singular Lane–Emden model (MFMS-LEM) by keeping the ideas of the literature LEM and by extension of the work of doubly singular multi-fractional LEM. This mathematical novel MFMS-LEM is numerically treated by applying the fractional Meyer neuro-evolution intelligent solver (FMNEICS). The optimization is performed using the mutual heuristics of fractional Mayer wavelet neural networks (FMW-NN), the global search aptitude of genetic algorithms (GAs) and interior-point algorithm (IPA), i.e., FMW-NN-GAIPA. The derivation steps, details of the singular points, fractional terms, shape factors and singular points are also provided. The modeling strength of MW-NN is implemented to characterize the novel model in the sagacity of mean squared error of objective function and network optimization is performed with the integrated capability of GAIPA. The authentication, perfection and verification of FMNEICS is checked for three diverse cases of the novel model which are conventional via relative studies through the reference solutions based on accuracy, stability, robustness and convergence procedures. Furthermore, the explanations via the statistical measures validate the value of the designed stochastic solver FMW-NN-GAIPA.
20 citations
References
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TL;DR: Goldberg's notion of nondominated sorting in GAs along with a niche and speciation method to find multiple Pareto-optimal points simultaneously are investigated and suggested to be extended to higher dimensional and more difficult multiobjective problems.
Abstract: In trying to solve multiobjective optimization problems, many traditional methods scalarize the objective vector into a single objective. In those cases, the obtained solution is highly sensitive to the weight vector used in the scalarization process and demands that the user have knowledge about the underlying problem. Moreover, in solving multiobjective problems, designers may be interested in a set of Pareto-optimal points, instead of a single point. Since genetic algorithms (GAs) work with a population of points, it seems natural to use GAs in multiobjective optimization problems to capture a number of solutions simultaneously. Although a vector evaluated GA (VEGA) has been implemented by Schaffer and has been tried to solve a number of multiobjective problems, the algorithm seems to have bias toward some regions. In this paper, we investigate Goldberg's notion of nondominated sorting in GAs along with a niche and speciation method to find multiple Pareto-optimal points simultaneously. The proof-of-principle results obtained on three problems used by Schaffer and others suggest that the proposed method can be extended to higher dimensional and more difficult multiobjective problems. A number of suggestions for extension and application of the algorithm are also discussed.
6,411 citations
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TL;DR: A reliable algorithm is employed to investigate the differential equations of Lane-Emden type using the Adomian decomposition method with an alternate framework designed to overcome the difficulty of the singular point.
360 citations
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TL;DR: In this paper, a perturbative technique for quantum field theory consists of replacing nonlinear terms in the Lagrangian such as φ4 by (φ2)1+δ and then treating δ as a small parameter.
Abstract: A recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the Lagrangian such as φ4 by (φ2)1+δ and then treating δ as a small parameter. It is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the Lane–Emden, Thomas–Fermi, Blasius, and Duffing equations.
301 citations
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TL;DR: In this paper, the general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated.
Abstract: The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter.
It is shown that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers.
257 citations
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TL;DR: In this paper, the general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated.
242 citations