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Reduction Method for the Solution of Weakly Singular Integro-Differential Equations

机译：弱奇异积分微分方程解的约简方法

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Approximation of functions of a complex variable by various finite-dimensional aggregates is an important problem not only in constructive function theory and approximation but also in the justification of direct approximate methods for functional equations. This problem has been well studied for the case of functions defined on standard contours (a straight line segment, the unit circle, and so on). In the case of an arbitrary closed smooth contour Γ in the complex plane, the problem is less studied. It should be noted that conformal mapping from the arbitrary smooth closed contours to the unit circle does not solve the problem. Moreover, it makes more difficulties: 1. The coefficients, kernel and right part of the transformed equation lose their smoothness; 2. The power of smoothness appears in convergence speed of collocation method. So that the evaluations of convergence speed will depend from particular contour; 3. The numerical schemes of researched methods become more difficult. The singularity appears in new kernel and we are not able to use the numerical schemes of mechanical quadrature method because of a singularity for new kernel. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for the approximate solution of weakly singular integro- differential equations defined on smooth closed contours in the complex plane. We use the cut-off technique kernel to reduce the weakly singular integro- differential equation to the continuous one. Our approach is based on the Krykunov theory and Zolotarevski results. We obtain the theoretical justification in Generalized Holder spaces.
机译：通过各种有限维集合对复杂变量的函数进行逼近，不仅在构造函数理论和逼近中，而且在对函数方程的直接逼近方法的证明中，都是重要的问题。对于在标准轮廓（直线段，单位圆等）上定义的功能的情况，已经对该问题进行了很好的研究。在复平面中任意闭合的平滑轮廓Γ的情况下，对该问题的研究较少。应当注意，从任意光滑的闭合轮廓到单位圆的保形映射不能解决该问题。而且，这带来了更多的困难：1.变换后的方程的系数，核和右部分失去了平滑度； 2.平滑的力量出现在配置方法的收敛速度上。因此，收敛速度的评估将取决于特定的轮廓； 3.研究方法的数值方案变得更加困难。奇异性出现在新内核中，由于新内核的奇异性，我们无法使用机械正交方法的数值方案。对于复杂平面上光滑闭合轮廓上定义的弱奇异积分微分方程的近似解，我们建议了Faber-Laurent多项式系统上的约简方法的数值方案。我们使用截止技术内核将弱奇异积分微分方程简化为连续方程。我们的方法基于Krykunov理论和Zolotarevski结果。我们获得了广义Holder空间中的理论证明。

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