Abstract
In the present research work, we designed a hybrid stochastic numerical solver to investigate nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions arising in various physical models. In this method, we hybridized Harris Hawks Optimizer with Interior Point Algorithm named HHO-IPA. We construct artificial neural networks (ANNs) model for the problem, and this model is tuned with the proposed scheme. This scheme overcomes the singular behavior of problems. The accuracy and applicability of the method are illustrated by finding absolute errors in the solution. The outcomes are compared with the results present in the literature to demonstrate the effectiveness and robustness of the scheme by considering four different nonlinear singular boundary value problems. Further, the convergence of the scheme is proved by performing computational complexity analysis. Moreover, the graphical overview of statistical analysis is added to our investigation to elaborate further on the scheme’s stability, accuracy, and consistency.
Highlights
The class of nonlinear singular differential equations performs a vital role in many fields of science and technology
The disadvantages related to the convergence of the series solution and computational complexity of the above listed numerical approximation algorithms motivated us to develop a stochastic algorithm based on artificial neural networks (ANNs) for the numerical treatment of the problem given in equation (1) subjected to two different kinds of boundary conditions given in (2)
ANNs use the computational or mathematical model which is used for processing available information is an interconnected set/group/collection of artificial neurons, in other words artificial neural networks are utilize to model the complex relationship among the input and output data
Summary
The class of nonlinear singular differential equations performs a vital role in many fields of science and technology. The main reason for designing AADM is to deal with nonlinear problems without considering unrealistic norms like discretization and linearization and improve the convergence as the author claimed in [26] To achieve this goal, the computational complexity of the algorithm increase. The disadvantages related to the convergence of the series solution and computational complexity of the above listed numerical approximation algorithms motivated us to develop a stochastic algorithm based on artificial neural networks (ANNs) for the numerical treatment of the problem given in equation (1) subjected to two different kinds of boundary conditions given in (2). The proposed algorithm is developed based on a minimizing function of mean square error in the differential equation together with mean square error in boundary conditions called fitness function to get the optimal solution of a given problem.
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