This paper addresses a computational technique for the numerical solutions of nonlinear distributed-order fractional boundary value equations with singular coefficients. The proposed strategy is based on the shifted Legendre-series expansion and the composite midpoint quadrature rule. Moreover, a collocation technique is utilized to reduce the understudy equations to a system of nonlinear algebraic equations solved by Newton’s iteration formula. The \(l_{2}\) and \(l_{\infty }\)-norm errors and experimental convergence order are selected as criteria to analyze the accuracy and precision of the proposed strategy. The results of the performed numerical simulations illustrate the reliability and validity of the proposed approach.

本文讨论了具有奇异系数的非线性分布阶分数边值方程的数值解的计算技术。所提出的策略基于移动的勒让德级数展开和复合中点求积法则。此外，利用搭配技术将待研究方程简化为由牛顿迭代公式求解的非线性代数方程组。选择\(l_{2}\)和\(l_{\infty }\) -范数误差和实验收敛阶数作为标准来分析所提出策略的准确性和精度。所执行的数值模拟的结果说明了所提出方法的可靠性和有效性。