All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.

中文翻译：

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具有不同奇异核的二次非线性积分方程的解

以前的所有作者仅通过不同方法讨论了具有连续核的二次方程。在本文中，我们引入了具有对数形式和Carleman型奇异核的混合非线性二次积分方程（MQNLIE）。讨论了MQNLIE的存在性和唯一性。应用正交方法获得非线性积分方程组（NLIE），然后使用Toeplitz矩阵方法（TMM）和Nystrom方法获得非线性代数系统（NLAS）。牛顿-拉夫森法用于求解获得的NLAS。在每种方法中，使用Maple 18软件考虑一些数值示例，并计算其估计误差。