1. ๏ฉ๏ฎ๏ด๏ฒ๏ฏ๏ค๏ต๏ฃ๏ด๏ฉ๏ฏ๏ฎ
Recently several studies (see e.g. references [1,2]) have been reported in which the solutions of both constant and time-varying systems are expressed in terms of Chebyshev polynomials. The first applications of orthogonal polynomials to differential equations with periodic coefficients were reported by Sinha and Chou [3] and Sinha et al. [4]. Although these applications were limited to second order scalar equations only, a later study by Sinha and Wu [5] presented a general scheme for the solution and stability analysis of a system of second order equations. The approach was based on the idea that the state vector and the periodic matrix of the system can be expanded in terms of Chebyshev polynomials over the principal period. After obtaining an equivalent set of integral equations and utilizing the product and integration operational matrices associated with the shifted Chebyshev polynomials of the first kind, such an expansion reduces the original problem to a set of linear algebraic equations from which the solution in the interval of one period can be obtained. Furthermore, the technique was combined with Floquet theory to yield the state transition matrix at the end of one period and thus provide stability conditions in addition to providing the solution after the first period. An error analysis concluded that the suggested schemes not only provide accurate results with rapid convergence, but are also computationally very efficient. In particular, the second order ''direct formulation'' was found to be several times faster than the standard Runge-Kutta type codes [6]. Whether applied to second or especially higher order equations, however, this formulation involves lengthy manipulations through integration by parts. Subsequently, a more straightforward and efficient ''differential formulation'' was proposed by Sinha and Butcher [7], which eliminated the need to integrate by parts since the original differential system was expanded directly. For this purpose, the differentiation operational matrix associated with the shifted Chebyshev polynomials was derived and used in place of the integration operational matrix. However, since the initial conditions did not enter as an integration constant, they had to be incorporated into the resulting algebraic system by replacing the individual equations which had lost the most accuracy due to differentiation of the Chebyshev series according to rules which changed depending on the size and structure of the problem.
Here, a new ''hybrid formulation'' is presented; first for a time-periodic pth order system (p arbitrary) and then specifically for a set of second order equations in which all system matrices are time-periodic, and with the aim of incorporating the advantages of both the ''integral'' and ''differential'' formulations. It will be shown that this formulation, which utilizes both the integration and differentiation operational matrices, eliminates both (1) the need for tedious integration by parts and (2) the need to insert the initial conditions