On Exact Solutions of a Class of Singular Partial Integro-Differential Equations

In this work for a class of second order model and non-model partial integro-differential equations with singularity in the kernel by the aid of arbitrary functions obtained integral representation for solution manifold. In this given article it is entered a new class of functions that at point (a, b) is converted to zero with some asymptotic behavior and the solution of given equation is found in considering class. Also, singular integro-differential operators are entered and main properties of these operators are learned. Also inverse operators for these operators are found. It is shown that the solution of studying equation is equivalent to the solution of system of two ordinary integro-differential equations of variables x and y. In the cases, when the solution of given integro-differential equation depends of any arbitrary constant Cauchy type problems are investigated. In order to study Cauchy type problems the properties of their solutions are studied. It is shown that, when some conditions are fulfilled the each of Cauchy type problems has only unique solution.