On a pseudo-Volterra nonhomogeneous integral equation

Abstract

In this paper the issues of the solvability of a pseudo-Volterra nonhomogeneous integral equation of the second kind are studied. The solution to the corresponding homogeneous equation and the classes of the uniqueness of the solution are found in [1]. By replacing the right - hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not «compressible». Using the Laplace transform, the obtained equation is reduced to an ordinary first - order differential equation (linear). Its solution is found. By using the solution of the homogeneous equation the form of a particular solution of the nonhomogeneous differential equation is defined (by the variation method of an arbitrary constant). By using the inverse Laplace transform, a particular solution of the pseudo-Volterra nonhomogeneous integral equation under study is obtained. The case of an nonhomogeneous integral equation with the value of the parameter k =1 is considered and studied. Classes for the right side and the solution of the integral equation are indicated.

Keywords

• pseudo-Volterra nonhomogeneous integral equation
• class of essentially bounded functions
• , inverse Laplace transformation
• residue