This paper presents a novel numerical algorithm that integrates reproducing kernel functions with Legendre polynomials to effectively address multiple interface problems. We create a new set of bases within the reproducing kernel space, introduce a linear operator, and utilize its properties to derive an equivalent operator equation. The model equation is then transformed into a matrix equation, enhancing the solution process for complex interface issues. Comparative numerical examples demonstrate the superior accuracy of our method over conventional approaches. Furthermore, the solution's existence and uniqueness are validated, ensuring the algorithm's reliability and effectiveness.