A distanced-network-based numerical approximation method for second-order differential equations

Ray-Ming Chen; Ray-Ming Chen

doi:10.3934/math.20261259

AIMS Mathematics

2026, Volume 11, Issue 9: 32020-32045. doi: 10.3934/math.20261259

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A distanced-network-based numerical approximation method for second-order differential equations

Ray-Ming Chen ^{1,2
,
,}

1.
Department of Mathematical Sciences, College of Science, Mathematics and Technology, Wenzhou-Kean University, Wenzhou 325060, China
2.
Department of Mathematical Sciences, College of Science, Mathematics and Technology, Kean University, 1000 Morris Avenue, Union, New Jersey 07083, USA

Received: 31 May 2026 Revised: 17 September 2026 Accepted: 22 September 2026 Published: 29 September 2026
MSC : 65L10, 65L60, 46B70

This research incorporates a distanced network into the calculus of variations to yield a distanced-network-based method (DNBM), which approximates and interpolates the solutions, or $ \hat{y}(x) $, for a boundary value problem (BVP) of second-order differential equations. Given an input vector $ X $ and boundary conditions $ y(x_0) $ and $ y(x_{N+1}) $, we identify the distance matrix $ D $ for $ X $ and choose a function $ \rho $ to reveal the relationship between the inputs, or $ D_{\rho} $. We perform calculus of variations to find the internal relationships between the variables and parameters. We also demonstrate how to apply the theory when $ D_{\rho} $ is singular by adding one weighted matrix to avert the singularity. A final experimental comparison between our DNBM and the Ritz finite-element method is also conducted. Our network method should provide another comprehensive and insightful approximation.
- metricized interpolation,
- calculus of variation,
- BVP,
- distanced network
Citation: Ray-Ming Chen. A distanced-network-based numerical approximation method for second-order differential equations[J]. AIMS Mathematics, 2026, 11(9): 32020-32045. doi: 10.3934/math.20261259

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Abstract

This research incorporates a distanced network into the calculus of variations to yield a distanced-network-based method (DNBM), which approximates and interpolates the solutions, or $ \hat{y}(x) $, for a boundary value problem (BVP) of second-order differential equations. Given an input vector $ X $ and boundary conditions $ y(x_0) $ and $ y(x_{N+1}) $, we identify the distance matrix $ D $ for $ X $ and choose a function $ \rho $ to reveal the relationship between the inputs, or $ D_{\rho} $. We perform calculus of variations to find the internal relationships between the variables and parameters. We also demonstrate how to apply the theory when $ D_{\rho} $ is singular by adding one weighted matrix to avert the singularity. A final experimental comparison between our DNBM and the Ritz finite-element method is also conducted. Our network method should provide another comprehensive and insightful approximation.

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