We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss–Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is first transformed into the interval [− 1, 1], by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss–Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme.

我们提出了一种基于广义Jacobi小波和高斯-雅各比正交公式的频谱配点方法，用于求解一类第三类Volterra积分方程。为此，首先考虑变量的适当变化，将积分间隔转换为间隔[-1，1]。然后，通过引入特殊的Jacobi参数，使用Gauss-Jacobi正交法则对积分部分进行近似。根据未知系数的雅可比小波函数考虑未知函数的近似，必须确定该系数。通过将该近似值代入方程式，并将所得方程式配置在一组搭配点处，可以获得线性代数方程式系统。然后，我们建议一种确定达到一定精度所需的基函数数量的方法。最后，包括一些示例以说明新方案的适用性，效率和准确性。