Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0,T). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L²- and \( L^{\infty } \)- norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper.

本文的目的是为两点边值问题的非线性系统开发一种Legendre-Jacobi配点方法，在间隔（0，T）上导数阶数最多为2 。该方案是基于将系统简化为Volterra-Fredholm积分方程的等效系统而构造的。在L ^2-和\（L ^ {\ infty} \）中都建立了该方法的频谱收敛速率。-规范。所得的光谱方法能够针对平滑问题和非平滑问题的合理收敛阶数实现光谱精度。而且，该方案易于数字实现。在各种复杂程度各异的问题上证明了该方法的适用性。就我们所知，这种分数阶微分方程非线性系统及其相关的Volterra-Fredholm积分方程非线性系统的频谱解尚未在文献中进行详细研究。本文填补了文献中的空白。