Abstract

In this paper for a class of model partial integro-differential equation with super singularity in the kernels, is obtained an integral representation of manifold solutions by arbitrary constants. The conjugate equation for the above-mentioned type of equations is also investigated. Such types of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of theory of analytical functions. However, the method of analytical functions is not applicable for our case of super singular equations with integrals understanding in Riemann–Stieltjes sense. Here, we have used the method of representation the considering equation as a product of two one-dimensional singular first order integro-differential operators. Further, a complete integro-differential equation and its conjugate equation have been investigated. It is shown that in every cases of characteristic equation roots the homogeneous integro-differential equation can have a nontrivial solutions. Non-model equation is investigated by the regularization method. Regularization of non-model equation is based on selecting a model part of equation. On the basis of the analysis of a model part of equation the solution of non-model equation reduced to the solution of a second kind Volterra integral equations with super singular kernel. It is important to emphasize that in contrast to the usual theory of Volterra integral equations, the studied homogeneous integral equation has nontrivial solutions. It is easy to see that the presence of a non-model part in the equation does not affect to the general structure of the obtained solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important.

中文翻译：

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新型超奇异积分微分方程及其共轭方程

摘要

本文针对一类具有超奇异性的局部偏微分方程模型，利用任意常数获得流形解的积分表示。还研究了上述类型方程的共轭方程。这种类型的积分微分方程不同于柯西型奇异积分微分方程。用解析函数理论的方法研究了柯西型奇异积分微分方程。但是，解析函数的方法不适用于我们在Riemann–Stieltjes意义上具有积分理解的超奇异方程组的情况。在这里，我们使用表示考虑方程的方法作为两个一维奇异一阶积分微分算子的乘积。进一步，研究了一个完整的积分微分方程及其共轭方程。结果表明，在每种情况下，特征方程根都是齐次的积分微分方程可以具有非平凡的解。通过正则化方法研究非模型方程。非模型方程的正则化基于选择方程的模型部分。在对方程模型部分进行分析的基础上，将非模型方程的解简化为具有超奇异核的第二类Volterra积分方程的解。需要强调的是，与通常的Volterra积分方程理论相反，所研究的齐次积分方程具有非平凡解。不难看出，方程中非模型部分的存在不会影响所获得解的总体结构。从这里开始，对于给定类别的积分微分方程，研究模型方程变得很重要。