Document Type : Research Article
Authors
Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran.
10.30473/coam.2025.73348.1281
Abstract
This study employs the radial basis function-generated finite difference (RBF-FD) method to address high-dimensional elliptic differential equations under Dirichlet boundary conditions. The method utilizes polyharmonic spline functions (PHSs) combined with polynomials for approximation. A notable benefit of this approach is that PHSs do not require a shape parameter, simplifying implementation and enhancing numerical stability. The proposed method offers several advantages, including high accuracy, rapid computation, and adaptability to complex geometries and irregular node arrangements. It is particularly effective for high-dimensional problems, providing a mesh-free alternative that scales efficiently with increased complexity. Beyond scientific computing, the method is also applied to financial option pricing, where integro-differential equations are transformed into a series of second-order elliptic partial differential equations (PDEs). Numerical experiments demonstrate that the proposed algorithm significantly outperforms existing RBF-based approaches in both accuracy and efficiency. These strengths make it a robust tool for solving a wide range of PDEs in both regular and irregular domains.
Highlights
- Develops a robust Radial Basis Function-Generated Finite Difference (RBF-FD) approach tailored for high-dimensional elliptic differential equations with Dirichlet boundary conditions.
- Employs polyharmonic splines (PHS) functions combined with polynomial, eliminating the need for shape parameter tuning and thereby simplifying implementation and improving numerical stability.
- Facilitates the efficient resolution of complex PDE problems, including the transformation of parabolic partial integro-differential equations into second-order elliptic PDEs for streamlined computation.
- Demonstrates high accuracy, robustness, and computational efficiency across both regular and irregular domains, ensuring scalability in high-dimensional settings.
- Offers a mesh-free, flexible framework that is adaptable to complex geometries and irregular node distributions, thereby mitigating some limitations associated with highly distorted computational domains.
- Highlights the potential for extensions to nonlinear problems and more sophisticated scenarios, such as biharmonic equations, through alternative RBFs, with scope for future enhancement via adaptive node refinement and hybrid RBF approaches.
Keywords
- Radial basis functions
- Polyharmonic splines
- High-dimensional partial differential equations
- Finite difference method
- Option pricing
Main Subjects
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