Application of the Fourier cosine series to the approximation of solutions to initial non-Dirichlet boundary-value problems

Abstract

The paper deals with an application of the Fourier cosine series to the determination of an approximate solution to some one-dimensional initial boundary-value problems. With the new approach one can approximate solutions of many equations of engineering and physics, without solving the eigenvalue problems. It has been found out that the new method can successfully be used for linear partial differential equations with non-Dirichlet boundary conditions. The heat equation and the wave equation with constant coefficients have been solved using the method described. The solutions have been compared to those obtained by means of the method of seperation of variables. The numerical results show that the new solutions approximate well the classical solutions. For the heat equation, even the boundary conditions at the initial instant of time are satisfied. This does not occur, however, in the case of the wave equation, since the initial displacement of the rod does not satisfy prescribed boundary conditions.