AN EXACT SOLUTION FOR DIFFERENTIAL EQUATION GOVERNING VIBRATIONS OF A CIRCULAR MEMBRANE

Abstract

Abstract

The focus of this paper is the application of powerful Adomian Decomposition method to find an exact solution for differential equation governing vibrations of a circular membrane. The generalized Fourier-Bessel series expansion is applied to obtain an exact solution. The obtained analytical solution is simplified in terms of a given orthogonal basis function that these functions satisfy the boundary conditions. The best Fourier-Bessel series expansion in the case of our physical problem is the set of Eigenfunctions of the self–adjoint system. Previous studies indicate that the Eigen value problem yields an infinite set of real Eigen values and Eigen functions. These functions constitute the basis for the infinite- dimensional Hilbert space.

Therefore, every function which is continuous satisfies the boundary conditions of the system that can be expanded in an absolutely and uniformly convergent series in the Eigen functions and this functions are normalized.

In this paper, the generalized Fourier-Bessel series expansion functions are determined that these functions satisfy the boundary conditions before and after applying the Lm operator to the functions which either are zero or do not satisfy the boundary conditions at all.

To prevent this difficulty the functions are expanded in terms of known orthogonal functions and these functions are selected to satisfythe boundary conditions before and after applying Lm operator. For the first time, we solved this equation using ADM and compared the result with those of classical methods to demonstrate the validity of the present study