New Type Super Singular Integro-Differential Equation and Its Conjugate Equation

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.