In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm. Towards this end, a recently advanced analytical approach is proposed to obtain approximate solutions of nonclassical-types boundary value problems of fractional derivatives in Caputo sense. This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions. Moreover, the solution is presented in the form of a fractional series over the entire Hilbert spaces without unwarranted assumptions on the considered models. The validity of the present algorithm is illustrated by expounding and testing two numerical examples. The achieved results indicate that the proposed algorithm is systematic, feasibility, stability, and convenient for dealing with other fractional systems emerging in the physical, technology and engineering.

本文利用迭代再现核算法讨论了分数阶微分方程与积分边界条件的耦合系统。为此，提出了一种最新的高级分析方法来获得Caputo意义上的分数阶导数的非经典类型边值问题的近似解。该方法基于Sobolev空间上的Gram-Schmidt过程优化了近似解，该过程执行以快速收敛速率生成傅里叶展开，从而构造的核函数满足齐次积分边界条件。此外，在整个希尔伯特空间上以分数序列的形式给出了解决方案，而对所考虑的模型则没有任何不必要的假设。通过阐述和测试两个数值例子说明了本算法的有效性。取得的结果表明，该算法具有系统性，可行性，稳定性，并且可以方便地处理物理，技术和工程领域出现的其他分数系统。