Barycentric rational interpolation collocation method for solving two-dimensional linear and nonlinear
Wei Zhang1,2, Liangli Chen1
1School of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, China.
Plos One
|July 29, 2026
Summary
This study introduces a high-precision barycentric rational interpolation collocation method for solving two-dimensional integro-differential equations (2D-IDEs). The method demonstrates excellent numerical stability and accuracy for both linear and nonlinear problems.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Scientific Computing
Background:
- Barycentric rational interpolation offers superior numerical stability, accuracy, and adaptability to node distribution.
- Solving two-dimensional integro-differential equations (2D-IDEs) is crucial in various scientific and engineering fields.
- Existing methods may face challenges with stability and accuracy for complex 2D-IDEs.
Purpose of the Study:
- To propose a novel high-precision barycentric rational interpolation collocation method for 2D-IDEs.
- To provide accurate approximate solutions for linear and nonlinear 2D-IDEs with boundary conditions.
- To analyze the error estimation and convergence of the proposed numerical method.
Main Methods:
- Developing a discrete scheme for 2D-IDEs using the two-dimensional barycentric rational interpolation formula.
- Employing the two-dimensional Gauss-Legendre numerical integration formula for accurate discretization.
- Implementing a collocation approach to solve the resulting system of equations.
Main Results:
- The proposed method achieves high precision and numerical stability for solving 2D-IDEs.
- Error estimation and convergence analysis confirm the method's theoretical underpinnings.
- Numerical examples validate the effectiveness and accuracy of the barycentric rational interpolation collocation method.
Conclusions:
- The developed high-precision barycentric rational interpolation collocation method is effective for 2D-IDEs.
- The method offers a robust and accurate approach for computational solutions in relevant scientific domains.
- Further applications of this technique to complex differential and integral equations are warranted.
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