A stochastic numerical approach for a class of singular singularly perturbed system
Zulqurnain Sabir1, Thongchai Botmart2, Muhammad Asif Zahoor Raja3
1Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan.
Plos One
|November 28, 2022
Summary
This study introduces a novel neuro-evolutionary scheme, ANNs-PSO-IPA, for solving singular singularly perturbed boundary value problems (SSP-BVPs). The method demonstrates accuracy and robustness in solving complex mathematical problems.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics
Background:
- Singularly perturbed boundary value problems (SP-BVPs) present significant challenges in numerical computation.
- Traditional methods often struggle with the boundary layers inherent in singular SP-BVPs.
- Efficient and accurate numerical schemes are crucial for analyzing these problems.
Purpose of the Study:
- To develop and present a novel neuro-evolutionary computational scheme for solving singular singularly perturbed boundary value problems (SSP-BVPs).
- To enhance the solution accuracy and robustness of existing numerical methods for SSP-BVPs.
- To validate the proposed scheme through rigorous testing on various SSP systems.
Main Methods:
- A hybrid neuro-evolutionary approach combining artificial neural networks (ANNs), particle swarm optimization (PSO), and an interior-point algorithm (IPA) was developed.
- An error-based fitness function was formulated using the differential equations and boundary conditions of the SSP-BVPs.
- The optimization of the fitness function was achieved through the integrated ANNs-PSO-IPA computational framework.
Main Results:
- The proposed ANNs-PSO-IPA scheme was tested on four distinct cases of two SSP systems, demonstrating effective performance.
- The accuracy of the scheme was confirmed by comparing the obtained solutions with exact analytical solutions.
- Performance indices and statistical analyses from 100 independent runs confirmed the scheme's convergence, robustness, and accuracy.
Conclusions:
- The developed ANNs-PSO-IPA neuro-evolutionary scheme provides an effective and reliable method for solving singular singularly perturbed boundary value problems.
- The hybrid approach successfully integrates global and local search strategies for robust optimization.
- The scheme's validated accuracy and robustness make it a valuable tool for computational mathematics and related scientific fields.
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