According to mathematical literature, the parametric spline functions have not been used directly for solving the integral equations so far. In this work, we develop the tension spline approximation to obtain the numerical solution of Volterra–Fredholm integral equation. First, the tension spline method was obtained and then the combination of this method with the quasi-linearization method has been used for solving the nonlinear Volterra–Fredholm integral equation. The error analysis of the method is given. Illustrative examples are included to verify the effectiveness and applicability of the presented approach.

中文翻译：

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张力样条求解非线性Volterra-Fredholm积分方程的迭代方法

根据数学文献，到目前为止，尚未将参数样条函数直接用于求解积分方程。在这项工作中，我们开发了张力样条逼近以获得Volterra-Fredholm积分方程的数值解。首先，获得了张力样条法，然后将该方法与准线性化方法结合起来，用于求解非线性Volterra-Fredholm积分方程。给出了该方法的误差分析。包括说明性示例以验证所提出方法的有效性和适用性。