New matrix method for analytical solution of linear ordinary differential equations
by
Sina Khorasani
Georgia Institute of Technology, School of Electrical and Computer Engineering, Atlanta, GA 30332-0250
Coauthors: Ali Adibi

We report a new analytical method, named differential transfer matrix method (DTMM) for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the the transfer matrix formalism and its extension into the limiting differential form. The approach reduces the nth-order differential equation to a system of n first order linear differential equations in terms of the n independent solutions. The full analytical solution can be then found either by the Dyson's perturbation technique or numerical integration.

We present a simple expression for calculation of the derivatives of functions up to the order n, and also prove the fundamental theorem of DTMM in support of the general validity of the method.

We discuss the general properties of differential transfer matrices and solutions for the case of periodic coefficients. We also present several analytical examples, showing the applicability of the method, with particular emphasis on second-order equations. We show that the Abel-Liouville-Ostogradski theorem can be easily recovered through this approach.

The main advantage of this approach compared to other methods is that the evolution of independent solutions are directly computed, rather than their derivatives. This leads therefore to a general method for extraction of physical properties from the mathematical model of the system.

Date received: March 1, 2003