We consider convolution integral equations on a finite interval with a real-valued kernel of even parity, a problem equivalent to finding a Wiener–Hopf factorization of a notoriously difficult class of 2 × 2 matrices. The kernel function is assumed to be sufficiently smooth and decaying for large values of the argument. Without loss of generality, we focus on a homogeneous equation and we propose methods to construct explicit asymptotic solutions when the interval size is large and small. The large interval method is based on a reduction of the original equation to an integro-differential equation on a half-line that can be asymptotically solved in a closed form. This provides an alternative to other asymptotic techniques that rely on fast (typically exponential) decay of the kernel function at infinity, which is not assumed here. We also consider the problem on a small interval and show that finding its asymptotic solution can be reduced to solving an ODE. In particular, approximate solutions could be constructed in terms of readily available special functions (prolate spheroidal harmonics). Numerical illustrations of the obtained results are provided and further extensions of both methods are discussed.

我们考虑有限间隔的卷积积分方程，其实数值为偶数奇偶校验，这个问题等效于找到一个非常困难的2×2矩阵类的Wiener-Hopf因式分解。对于较大的参数值，假设内核函数足够平滑且衰减。在不失一般性的前提下，我们集中在一个齐次方程上，并提出了在区间大小较大时构造显式渐近解的方法。大间隔方法基于将原始方程式简化为半线上的积分微分方程式，该方程式可以渐近地以封闭形式求解。这提供了其他渐近技术的替代方法，这些渐进技术依赖于无限大时内核函数的快速（典型指数）衰减，在此不做假设。我们还以较小的时间间隔考虑了该问题，并表明找到其渐近解可以简化为ODE的求解。特别是，可以根据容易获得的特殊功能（近似球谐函数）构造近似解。提供了所得结果的数值说明，并讨论了这两种方法的进一步扩展。