Abstract This paper develops an accurate computational method based on the shifted Chebyshev cardinal functions (CCFs) for a new class of systems of nonlinear variable-order fractional quadratic integral equations (QIEs). In this way, a new operational matrix (OM) of variable-order fractional integration is obtained for these cardinal functions. In the proposed method, the unknown functions of a system of nonlinear variable-order fractional QIEs are approximated by the shifted CCFs with undetermined coefficients. Then, these approximations are substituted into the system. Next, the OM of variable-order fractional integration and the cardinal property of the shifted CCFs are utilized to reduce the system into an equivalent system of nonlinear algebraic equations. Finally, by solving this algebraic system an approximate solution for the problem is obtained. The main idea behind this approach is to reduce such problems to solving systems of nonlinear algebraic equations, which greatly simplifies the problem. Convergence of the presented method is investigated theoretically and numerically. Furthermore, the proposed approach is numerically evaluated by solving some test problems.

中文翻译：

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一类非线性变阶分数二次积分方程组的一种计算方法

摘要 本文针对一类新的非线性变阶分数二次积分方程 (QIE) 系统开发了一种基于平移切比雪夫基数函数 (CCF) 的精确计算方法。通过这种方式，为这些基数函数获得了一个新的变阶分数积分运算矩阵（OM）。在所提出的方法中，非线性可变阶分数 QIE 系统的未知函数由具有不确定系数的移位 CCF 逼近。然后，将这些近似值代入系统中。接下来，利用变阶分数积分的 OM 和移位 CCF 的基数性质，将系统简化为非线性代数方程的等效系统。最后，通过求解这个代数系统，可以得到问题的近似解。这种方法背后的主要思想是将此类问题简化为求解非线性代数方程组，从而大大简化了问题。从理论上和数值上研究了所提出方法的收敛性。此外，通过解决一些测试问题对所提出的方法进行了数值评估。