In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.

中文翻译：

---

求解非局部条件抛物型偏微分方程的再生核方法

在这项研究中，解决了具有非局部条件的抛物型偏微分方程。为此，我们使用了结合Galerkin方法的基本概念而获得的再生内核方法（RKM），以及Wang等人首先引入的完整的再生内核希尔伯特空间系统。他们在没有Gram–Schmidt正交化过程的情况下实施了RKM。在这种方法中，首先构造了满足非局部条件的可再生核空间及其核，然后对所考虑的问题实施了不使用Gram–Schmidt正交化过程的RKM。此外，还详细提供了收敛定理，误差分析定理和稳定性定理。为了显示本方法的高精度，解决了几个数值示例。