The problem of the vibration of a finite string with nonlinear perturbation (Q1589015)

This paper deals with the mixed problem to the finite string equation with the nonlinear perturbation \(f(t,x,u_t,u_x)\). The author reduces the problem under consideration to the solvability of an integro-differential equation whose kernel is the Green's function of a second-order ordinary differential equation with non-selfadjoint boundary conditions. The latter equation is equivalent to an appropriate system of nonlinear integro-differential equations that can be solved by the method of successive approximations. By this way the mixed problem to the string equation possesses a unique classical solution. Moreover, the solution can be represented in the form of a uniformly convergent Fourier series in eigenfunctions of the non-selfadjoint spectral problem mentioned above.