Spectral and pseudo-spectral Galerkin techniques, by using the standard Jacobi polynomials, are implemented to calculate numerically the solutions of pantograph type Volterra delay integro-differential equations that have kernels with the property of weak singularity. Because of the complex structure of the considered problems, pseudo-spectral Galerkin approaches are more desirable with respect to the spectral Galerkin approaches, since they have the property of integral approximator by using high order convergent Gauss quadrature formulas. A deep and detailed analysis of convergence of the numerical solutions to the exact solutions are given under some mild conditions such as smoothness of the solutions. Some test problems are illustrated and efficiency of the suggested numerical approach is investigated with respect to a recently proposed Jacobi pseudo-spectral collocation technique via some figures and tables experimentally.

通过使用标准的Jacobi多项式，使用光谱和伪光谱Galerkin技术来数值计算受电弓类型的Volterra延迟积分微分方程的解，该方程具有具有弱奇异性的核。由于所考虑问题的复杂结构，伪谱Galerkin方法相对于频谱Galerkin方法更为可取，因为它们具有使用高阶收敛高斯正交公式的积分逼近器的性质。在某些温和的条件下（例如平滑度），对数值解与精确解的收敛性进行了深入而详细的分析。