Abstract
The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation and the detailed discussions of the singularity, shape factor terms along with the fractional order forms. The numerical discussions of the FOPSS are described based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The FMWs-NN strength is pragmatic and forms a merit function based on the differential system and the initial conditions of the FOPSS. The merit function is optimized, using the integrated capability of PSOIPA. The perfection, verification and substantiation of the FOPSS using the FMWs is pragmatic for three cases through relative investigations from the true results in terms of stability and convergence. Additionally, the statics’ descriptions further authorize the presentation of the FMWs-NN-PSOIPA in terms of reliability and accuracy.
Highlights
The stochastic schemes based on numerical measures is applied to solve a variety of applications [33,34,35,36,37,38,39,40], and a few potential recently reported applications include the solution of nonlinear Lane–Emden multi-pantograph delay based ordinary differential equations (ODEs) [41], Gudermannian neural networks for sODEs [42], neuroswarming approach to singular with multiple delay ODEss [43], intelligent backpropagated networks for solving Lene–Emden singular ordinary differential systems with pantograph delays [44], novel design of Morlet wavelet neural networks for solving singular pantograph nonlinear differential models [45], third kind of multi-singular nonlinear systems [46], novel design of evolutionary integrated heuristics for singular systems [47], Morlet wavelet neural networks for solving higher order singular nonlinear ODEs [48] and wavelet analysis on some surfaces of revolution [49]
All these applications inspire the authors to investigate the design of fractional order pantograph singular system (FOPSS), which has never been implemented nor treated, by using the proposed heuristics of fractional Meyer wavelets (FMWs)-neural network (NN)-PSOIPA
The necessary comparison of the proposed FMWs-NN-PSOIPA is conducted with respect to magnitudes of variance account for (VAF), Theil’s inequality coefficient (TIC) and NSE for perfect modeling scenarios with values 100, 0 and 1, respectively
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The stochastic schemes based on numerical measures is applied to solve a variety of applications [33,34,35,36,37,38,39,40], and a few potential recently reported applications include the solution of nonlinear Lane–Emden multi-pantograph delay based ordinary differential equations (ODEs) [41], Gudermannian neural networks for sODEs [42], neuroswarming approach to singular with multiple delay ODEss [43], intelligent backpropagated networks for solving Lene–Emden singular ordinary differential systems with pantograph delays [44], novel design of Morlet wavelet neural networks for solving singular pantograph nonlinear differential models [45], third kind of multi-singular nonlinear systems [46], novel design of evolutionary integrated heuristics for singular systems [47], Morlet wavelet neural networks for solving higher order singular nonlinear ODEs [48] and wavelet analysis on some surfaces of revolution [49] All these applications inspire the authors to investigate the design of FOPSS, which has never been implemented nor treated, by using the proposed heuristics of FMWs-NN-PSOIPA.
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