Analytical and Numerical Bounds for a Nonlinear Overlap Model of Circular Sectors

Document Type : Research Article

Authors

Department of Mathematics, Lorestan University P. O. Box 465, Khoramabad, Iran

Abstract

In this paper, we investigate a nonlinear analytical model for estimating the overlap area between two circular sectors. The overlap problem is formulated in a functional–analytic framework by representing sector regions through indicator functions and interpreting the overlap area as a nonlinear trace expression involving multiplication operators on  $L^{2}(\mathbb{R}^{2})$. This formulation allows the application of classical nonlinear inequalities, including Young’s inequality and related operator bounds, to derive explicit analytical estimates for the overlap area. The resulting bounds depend nonlinearly on the sector parameters, such as angular widths and radii, and avoid direct geometric intersection computations. In addition, the proposed bounds are numerically tractable and can be efficiently evaluated numerically for a wide range of sector configurations. An angular–averaged nonlinear bound is introduced, providing a computable upper estimate that captures the combined angular and radial effects of the sectors. Several illustrative examples demonstrate the effectiveness of the analytical bounds and confirm their numerical consistency. The proposed approach establishes a connection between nonlinear analytical techniques, operator inequalities, and geometric modeling, offering a flexible framework for nonlinear overlap estimation problems. This formulation presents an innovative conceptual reformulation using operator theory for the circular overlap problem, providing a powerful tool for analysis and accurate estimation of the overlap area using classical inequalities.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 1
June 2026
Pages 142-152

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History

  • Receive Date: 16 February 2026
  • Revise Date: 30 March 2026
  • Accept Date: 06 April 2026
  • Publish Date: 07 June 2026

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