In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

中文翻译：

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具有泛函参数的李卡提方程的勒让德搭配法

在这项研究中，提出了一种在混合条件下数值求解具有泛函参数的 Riccatti 型微分方程的方法。该方法将勒让德多项式、解形式和所需表达式写成矩阵形式，并定义了配置点。然后，利用得到的矩阵关系和配置点，将Riccati问题简化为非线性代数方程组。问题中的条件写成矩阵形式，并借助得到的矩阵关系找到了一个新的非线性代数方程组。该系统被求解，因此系数矩阵被检测到。该系数矩阵以解的形式写入，因此获得了近似解。此外，通过定义残差函数，建立了一个误差问题，并获得了给出更好数值结果的近似解。为了证明该方法的可信度和方便性，通过数值例子说明了所提出的方法和误差估计技术。因此，数值结果在表格和图表的帮助下更清楚地显示出来，并将结果与​​其他方法的结果进行了比较。