On Some Applications of Kontorovich–Lebedev Transforms

Abstract

The application of the Kontorovich–Lebedev integral transforms and dual integral equations to the solution of some mixed boundary value problems is considered. We reduce the diffusion and elastic problems to the solution of the proper mixed boundary value problem for the Helmholtz equation.The solution of this problem as derived by Lebedev is determined in the form of the Kontorovich–Lebedev integral transform from the solution of dual integral equation with modified Bessel function of pure imaginary order in the kernel.It is shown that we can resolve the above-mentioned problem for the Helmholtz equation in the form of single quadrature from the solution of the Fredholm integral equation. The dimension of the problem is lowered on unit by this, which is the essential advantage of this method. The examples permitting the complete analytical solution of the problem are given.