Abstract A generalized collocation method for solving a weakly singular Fredholm integro-differential equation with Kalman kernel is proposed in reproducing kernel space. To obtain the generalized collocation method, the multiwaves basis in reproducing kernel space W n + 1 [ 0 , b ] is constructed based on Legendre multiwaves in L 2 [ 0 , 1 ] . Using the multiwaves basis, we propose e-approximate solutions and use the method of searching the minimum to obtain the best approximate solution of the equation. Meanwhile, convergence order and stability of the generalized collocation method are studied. It is worth to show that the generalized collocation method proposed in the paper is stable and could be applied to solve other integral equations or differential equations.

中文翻译：

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求解弱奇异Fredholm积分微分方程的核空间再现广义搭配方法

摘要 在再现核空间中，提出了一种用卡尔曼核求解弱奇异Fredholm积分微分方程的广义搭配方法。为了得到广义的搭配方法，在L 2 [ 0 , 1 ] 中的Legendre 多波的基础上构造了再生核空间W n + 1 [ 0 , b ] 中的多波基。使用多波基，我们提出了e-近似解，并使用搜索最小值的方法来获得方程的最佳近似解。同时研究了广义搭配方法的收敛阶次和稳定性。值得证明的是，本文提出的广义搭配方法是稳定的，可以应用于其他积分方程或微分方程的求解。