The scaled boundary finite-element method: analytical solution in frequency domain Original Research Article

The scaled boundary finite-element equation, a system of ordinary differential equations in the radial coordinate, is solved analytically applying a series expansion. An eigenvalue problem of a Hamiltonian matrix is solved. When the difference of two eigenvalues is an even number, besides the power series, logarithmic functions occur. Two sets of independent solutions are obtained. The scaled boundary finite-element method is thus a semi-analytical procedure to solve partial differential equations with the polynomial approximation of finite elements in the two circumferential directions and a closed-form solution in the radial direction. Applications to bounded and unbounded domains demonstrate excellent accuracy.