This paper presents a polynomial-based reproducing kernel method (RKM) for approximating solutions to two-point boundary value problems. The approach constructs a kernel function using shifted Chebyshev polynomials, yielding solutions in the form of a truncated series. A key contribution is mitigating the dependence of convergence on dense nodal discretization—a common limitation in traditional RKMs. Numerical results demonstrate that the proposed method achieves comparable or superior accuracy with fewer nodes, significantly lowering computational costs.
In this study, we developed a Chebyshev-based reproducing kernel method (C-RKM) for efficiently solving two-point boundary value problems. The proposed approach retains the essential theoretical strengths of classical reproducing kernel methods while addressing two common limitations: the heavy dependence on dense nodal distributions and the computational cost associated with repeated orthogonalization.
By constructing the reproducing kernel using shifted Chebyshev polynomials that satisfy boundary conditions intrinsically, we established a dynamic solution space that evolves with the polynomial degree $m$. This flexibility allows for accurate approximations using relatively few nodal points, without compromising convergence or stability.
Numerical experiments confirmed the theoretical advantages of C-RKM. Compared to conventional RKM techniques, our method consistently required fewer nodes and yielded significantly better accuracy, particularly in problems featuring singularities or internal layers. The rapid error decay and low computational overhead make C-RKM a promising tool for practical applications in applied mathematics and engineering.
Future work may explore extending this framework to higher-order boundary value problems, systems of equations, or multi-dimensional domains using tensor-product Chebyshev kernels.