The nonlocal problem for a hyperbolic equation with bessel operator in a rectangular domain

Abstract

We consider a boundary value problem for a hyperbolic equation with Bessel differential operator in a rectangular domain with integral nonlocal boundary value condition of the first kind. The equivalence between boundary value problem with integral nonlocal condition of the first kind and a local boundary value problem with mixed boundary conditions of the first and third kinds is proved. The existence and uniqueness of solution of the equivalent problem are established by means of the spectral method. At the uniqueness proof the completeness of the eigenfunction system of the spectral problem is used . At the existence proof the assessment of coefficients of series, the asymptotic formula for Bessel function of the first kind and asymptotic formula for eigenvalues are used. Sufficient conditions on the functions defining initial data of the problem are received. The solution of the problem is obtained in explicit form. The solution is obtained in the form of the Fourier-Bessel series. Its convergence is proved in the class of regular solutions.

Keywords

• hyperbolic equation
• singular coefficient
• bessel differential operator
• non-local boundary value condition
• uniqueness
• existence
• fourier-bessel series
• uniform convergence