This paper is devoted to solve a class of weakly singular Volterra integro-differential equations with noncompact kernels. Since the solution of such these models may has singularities, so the classical spectral methods lose their high accuracy for solving them. Innovation of this article is that, we use the generalized mapped nodal Laguerre spectral collocation method to deal with the singularity. Therefore, we can make good use of the advantages of the mapped Laguerre functions. The best advantage of the proposed method is its robustness for solving problems that have singularity near the left (or right) boundary of the computational interval. We present the construction and analysis of the generalized log orthogonal Laguerre functions collocation method in this paper and some numerical examples with nonsmooth solutions are included to show the efficiency of the suggested numerical scheme with respect to the classical Jacobi spectral methods.

本文致力于解决一类具有非紧实核的弱奇异Volterra积分微分方程。由于这些模型的解可能具有奇异性，因此经典频谱方法失去了求解它们的高精度。本文的创新之处在于，我们使用广义映射节点的拉盖尔谱配点方法来处理奇点。因此，我们可以充分利用映射的Laguerre函数的优点。所提出的方法的最大优点是它对于解决在计算间隔的左（或右）边界附近具有奇异性的问题的鲁棒性。