Spectral Analysis of Matrix Differential Operators Using an Operational Matrix Method

Abstract

Analysis of the spectral of matrix differential operators is key to representing and understanding many coupled and multi-physics systems. Nevertheless, the eigenvalues and eigenfunctions for such operators are difficult to compute directly and efficiently in terms of traditional numerical methods (which are expensive and highly inaccurate). We suggest an effective spectral algorithm for the investigation of matrix differential operators through operational rules using Chebyshev polynomials. The method represents the unknown vector functions by Chebyshev series expansions and makes use of the derivative matrices to reduce a continuous eigenvalue problem to a standard algebraic matrix formulation. The resulting transformation allows to compute directly spectral properties with standard numerical solvers. It is shown that the proposed method has high accuracy, fast convergency and low computational cost in terms of a 2 × 2 and a series example with 3×3 matrix differential operators. It is demonstrated that the operational matrix-based spectral method is accurate and convenient to solve eigenvalue problems of matrix differential operators, which can be further generalized to a wide class of linear operator eigenvalue problem.